The second-half substitute, who had scored in his previous two midweek outings, slotted home after much-maligned goalkeeper Simon Mignolet had saved Stoke defender Marc Muniesa's effort in sudden death.
There was some irony the decisive moment was produced by one of the smallest players on the pitch in a match which manager Jurgen Klopp had predicted just before kick-off would be "a heading festival".
The ball certainly saw some air time but four shots on target in 120 minutes underlined the lack of quality in the game and penalties were almost inevitable from the moment Marko Arnautovic struck a controversial offside goal in the first half.
However, Liverpool extended to 42 occasions their proud record of never having been knocked out when returning to Anfield with a first-leg lead.
They have also never lost the home leg of a League Cup semi-final and have reached a record-extending 11th final, from which they have returned victorious eight times.
Stoke have still to win at Anfield since 1959, which includes 31 defeats and five draws, but this was just about as close as you could get.
They fielded five outfield players at 6ft-plus to Liverpool's two and it seemed they would target Liverpool's weakness at set-pieces.
Former Reds striker Peter Crouch, head and shoulders above his markers at 6ft 7ins, won virtually everything in the air but Liverpool, with Lucas Leiva at the forefront, did a reasonable job of mopping up the second ball.
But the hosts were undone seconds before the interval with a move through the middle of the pitch which picked out Bojan Krkic and, with left-back Alberto Moreno nowhere to be seen, he slid a pass across to the unmarked Arnautovic who side-footed home despite replays indicating he was offside.
It was the first shot on target in the game and the sixth time in 2016 Liverpool had conceded from such.
Jon Walters and Bojan both had chances for the visitors but Liverpool, holding a 1-0 lead going into the second leg, seemed reluctant to shoot with Adam Lallana and James Milner both guilty of passing inside the penalty area, although Emre Can and Jordan Henderson had attempts from distance.
Early in the second half Roberto Firmino forced Jack Butland to tip his shot onto the near post with his foot but Stoke quickly gained control and exerted considerable pressure, with Mamadou Sakho throwing himself at a Walters shot an example of the desperation.
Striker Christian Benteke replaced Henderson but it was Sakho who wasted a good Milner free-kick by heading wide, with Benteke failing to meet Moreno's cross under pressure from Erik Pieters.
The hosts managed just one shot on target as the match went into extra time, where substitute Marco van Ginkel hit the post just before the end of the first period and forced Simon Mignolet into a save in the second whereas Liverpool failed to test Butland.
Crouch and Germany international Emre Can both failed from the spot - but with Walters, Glenn Whelan, Ibrahim Afellay, Xherdan Shaqiri and Van Ginkel scoring for Stoke and Lallana, Benteke, Firmino, Milner and Lucas on target for the hosts, Allen struck after Muniesa failed.
Milner: Winning trophy can right Liverpool's season
Liverpool vice-captain James Milner believes a trophy will prove this season has not been written off because of a change of manager.
The replacement of Brendan Rodgers with Jurgen Klopp was supposed to herald a new era at Anfield but the German has found transforming the club's fortunes in the short term a difficult task and they are well off the pace for a Champions League place.
But ahead of a Capital One Cup semi-final second leg against Stoke, which they hold a 1-0 lead from the game at the Britannia, Milner insists if they can win silverware it will at least prove the players have not taken their eye off the ball with the changes which have happened.
"It's not about writing off this season, saying 'We've changed the manager, there's a lot of changes and let's go again next year'," he said.
"If we can win at least one trophy while changing manager and improving and new players coming in during a year of transition then we'll be achieving something and that's the aim.
"I think there are improvements to be made and we know where we need to get to.
"We're in a position where we haven't played as well as we would have liked and there's a lot of improvements to be made so we can move on.
"It's always nice to go to Wembley but we've got this job to do first."
Liverpool were the better team at the Britannia three weeks ago as Stoke struggling to find the footballing fluency Mark Hughes has instilled in them which has already seen them comfortably beat Manchester City and Chelsea at home.
It is now widely accepted they are a much-changed side from the team which played long-ball tactics and looked to capitalise on set-pieces - which is a shame for the Potters as Liverpool's primary weakness is from dead-ball situations.
As a result Liverpool need to find different ways to keep their opponents at bay.
"We used to put advertising boards around the Etihad Stadium to stop the throw-ins and the run up," said the former Manchester City midfielder.
"You don't have to do that any more but they're a top team now.
They have a lot of good players now and that's no disrespect to how they used to play.
"Nobody liked playing against them and they played a certain way but they've changed the style, a new manager has come and they've got a lot of quality players.
"It's down to us to finish the job."
Mark Hughes: Grasp historic opportunity
Stoke boss Mark Hughes has urged his players to grasp the historic opportunity that lies ahead of them in Tuesday's Capital One Cup semi-final second leg at Liverpool.
The Potters head into the contest trailing 1-0 from the first leg at the Britannia Stadium.
If they can turn the tie around, the Staffordshire outfit would be in a League Cup final for the first time since 1971-72, the season in which they won the competition to secure what is the only major trophy in their history.
And Hughes said on Monday: "It has been 40-odd years, and we remind the players of that because this is an opportunity to make your mark in the game.
"When that presents itself, you really have to grasp it.
"It is an opportunity we are not going to allow pass by lightly.
"We are going to give it everything - it is a chance for everybody involved here to achieve something that hasn't been achieved for many years.
"And it is important to us because if we can get to a final, win a trophy, then that gives you a winning mentality - you've experienced that as a group and that will help you to continue to progress.
"It is a big part of what we want to do moving forward."
Hughes admits it is a "big test" that awaits Stoke, whose last victory at Anfield came in 1959.
But he believes his side will benefit from the fact that they are in a must-win scenario.
"We are behind in the tie and we know exactly what we have to do," the Welshman said.
"There is no grey area for us - we have to win. And I think when you are in that situation as a player or a manager, it clears the mind.
"The clarity of what we need to do is going to be an advantage I think."
The match pits Stoke, currently ninth in the Premier League, against a team two places and one point better off than them in the table.
As well as losing the first leg, the Potters were also beaten by Liverpool - then managed by Reds boss Jurgen Klopp's predecessor Brendan Rodgers - in the league earlier this season.
But they thrashed them 6-1 at the end of the previous campaign, and have registered notable victories since against the likes of Chelsea, Manchester City and Manchester United.
Hughes said: "We are going to really push this.
"If we do what we're good at, we'll run them very close."
Stoke will be missing skipper Ryan Shawcross, with it being revealed on Monday that the defender is set to be sidelined for up to a month by the back injury he sustained in Saturday's 3-0 loss at Leicester.
Hughes admitted the news was a blow, but said: "I've still got good players who can come in and they will be prepared and ready to go."
Marko Arnautovic, who has been carrying a hamstring problem, was not part the squad at the weekend and fellow forward Bojan Krkic was an unused substitute, but both are expected to return to the starting line-up.
Geoff Cameron (ankle) remains sidelined but may be in contention for Saturday's FA Cup game at Crystal Palace.
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