Celtic boss Neil Lennon has congratulated Morton but refused to blast his players after the Hoops failed to secure a place in the Scottish League Cup quarter-finals.
Dougie Imrie's penalty in extra time saw the Championship strugglers book their place in the last eight.
The Parkhead giants laid siege to the visitors' goal but could find no way past the Greenock side's goalkeeper Nicolas Caraux.
"It's easy to criticise the players and we did everything but score," Lennon said.
"We had 26 corners, hit the post and had one cleared off the line. I've got to give credit to Morton who defended heroically at times.
"Did we do enough to win the game?
"Yes, but we didn't take our chances so, congratulations to Morton.
"That's an eye-opener for them again but some of them have to do better.
"They didn't have great height. Virgil van Dijk won everything and we couldn't capitalise on the umpteen headers he won.
"It's our bad luck and Morton's good luck," he continued.
"We've had a very demanding start to the season but we should have passed the ball better.
"I can't over-analyse it, we just didn't make the most of the chances that we created.
"I've got complete faith in the players and you think they'll eventually break them down and get one.
"We did everything and tried everything to score.
"It seems to be a little bit of a bogey competition.
"This is the first time we haven't made the semi-final for a long time; it's a bit sore but I'll get over it."
It was a magical moment for Morton as it was their first win at the home of the Scottish champions since 1971.
"We said: 'If you swap a jersey, make sure it's full of sweat so they know you have put a shift in'," commented Moore.
"I hope we can keep these standards up and push on in the league because that is our bread and butter.
"You need boys with desire and work-rate and they had it in abundance."
The fairytale triumph was Morton's first victory over Celtic since 1985.
Moore praised the Cappielow chairman Douglas Rae for his patience while the team has struggled this season and said he hoped the League Cup result would "be the catalyst for picking up points" in the league.
"I hope we can keep these standards up and push on in the league because that is our bread and butter," he added.