In the last five months, Brendan Rodgers' team have demonstrated a series of weaknesses that their manager has struggled to fix, but at least last night he did not have to add a lack of nerve against Championship opposition to the list.
It was a win at last for Rodgers, the first in four games, and some respite from the grip of anxiety that has taken hold at Anfield. For Raheem Sterling, without a goal in 17 games, there was two in this cup tie, and one for Lazar Markovic, although there was still a brief passage when Liverpool began to wobble.
There was no repeat of the Selhurst Park capitulation of last season, and Liverpool will face Chelsea in the semi-finals of the Capital One Cup. Bournemouth had their chances in the first half to keep the game within reach but they simply gave their Premier League opposition too much time and space.
Liverpool's first goal was a 52-pass move, beautifully constructed and the kind of football that Rodgers was hoping to play before the dream went sour.
Yet when Dan Gosling, the former Everton midfielder, scored to make it 3-1 before the hour there was a period when another goal might have seen Liverpool's will dissolve again. This time, they managed to steady the ship.
Rodgers picked a side for the occasion with just three changes from the team that lost at Old Trafford on Sunday. There was no resting captain Steven Gerrard, or Dejan Lovren and Martin Skrtel, the defensive axis which has been the source of so many problems this season.
There was also no backing down from the three-man defensive configuration either, with Kolo Toure taking the place of the injured Glen Johnson and within the first four minutes it felt possible that Liverpool might pay a high price for their manager's stubbornness.
Callum Wilson, Bournemouth's top goalscorer, picked his way past Lovren and Skrtel and, with just Brad Jones to beat, passed the ball wide of the goal.
It was a dreadful miss and by the time Bournemouth got sight of Jones' goal again, they were two goals behind.
What followed the Wilson chance was a period of Liverpool domination that felt so absolute that it was hard to recall a time when Bournemouth put a challenge on the away side's back three. Toure, Skrtel and Lovren were able to pass the ball under no pressure at all, and the Championship side barely had a touch.
It looked very much like Eddie Howe's team had been told to stand off Liverpool and try to absorb the attacks they made closer to the home goal.
The consequence was that Gerrard, Adam Lallana and Philippe Coutinho also passed the ball under little or no pressure and a Liverpool team with ebbing confidence was allowed to work its way into the game.
Liverpool were allowed to find their tempo with Coutinho and Lallana playing behind Sterling. On the wings, Markovic and Jordan Henderson offered the width. Their first goal was testament to the time and space they were afforded.
It ended with Henderson heading Markovic's cross from the left across goal for Sterling to nod past Artur Boruc. The move comprised of 52 passes and lasted two minutes and 26 seconds. There is no doubting that this Liverpool side, at their best, can pass the ball well.
Before then, Boruc had done well to push a Lallana shot over the bar. The goalkeeper had also stopped Gerrard's strong strike from Markovic's cutback but it felt like only a matter of time before Liverpool scored.
They did so again on 27 minutes when Markovic struck a low shot past Boruc after Coutinho's shot was saved. In the build-up, Bournemouth captain Tommy Elphick appeared to stop in response to what he seemed to think was the referee's whistle.
Then, a moment of hope for Howe's team. Right-back Simon Francis got away down the right side and cut the ball back out of the reach of Toure to Yann Kermorgant in the area. Unmarked and with time to pick his spot, he struck wildly over the bar.
For so many of Howe's Championship-leading team, it was to be one of those nights when they were reminded of what might lay ahead should they make the top division. Kermorgant was off before the hour and by then Liverpool had scored a third.
It came through the centre, from Toure to Lallana and on to Sterling. He slotted the ball beyond the reach of Boruc and if it had looked hard for Bournemouth before then, at three goals it was starting to seem impossible.
Yet they pulled one back before the hour, a goal from Gosling. His shot found its way under the body of Jones, newly-installed as the club's No 1, and doing little to convince that he is deserving of the position.
The goal alone showed how easily Liverpool's rhythm could be disrupted and for the next few minutes there was a shakiness about them.
One of the Bournemouth substitutes, Ryan Fraser, headed a chance wide. Gosling hit the post on 69 minutes.
It proved too much for Bournemouth to close the gap and Sterling had two good chances for the hat-trick.
By the end, Rodgers' name was being sung by the away support and, given the week they have had, he will certainly settle for that.