The former Hamilton and St Mirren player believed a return game at Cappielow was on the cards after he had helped keep the Hoops at bay for an hour and a half until he was told there would be 30 minutes added time.
Five minutes into the extra period, Imrie scored from the spot after referee Bobby Madden adjudged Celtic defender Efe Ambrose to have handled when challenging Greenock substitute Archie Campbell and the Scottish Championship side held out to book an unlikely quarter-final spot.
Afterwards he said: "We did really well, stuck to our task and got a penalty in extra time.
"Before that, I thought the game was over. I thought it was a replay.
"I was away clapping the fans at 90 minutes and was told there would be another 30 minutes.
"I was already knackered by then but it was great to get a win."
A delighted Morton boss Allan Moore revealed he had gone into the game on the back of a meeting with chairman Douglas Rae about his future at the club, after his side had slumped to second-bottom of the table.
He said: "I'd had a meeting with the chairman yesterday (Monday) because there were stories saying I was under pressure.
"He wanted to clarify it with me, I didn't seek to clarify it with him.
"He told me my job was 100 percent safe at the moment.
"He had never indicated my job was under pressure. He knows we've brought some young boys in.
"I don't need anyone to put pressure on me, I put pressure on myself.
"I can understand Morton fans' frustrations.
"It's a work in progress and hopefully we get time to bring that through."
Hoops manager Neil Lennon spoke before the game of his desire to win the competition for the first time as a manager but he was "philosophical" in defeat.
He said: "We should be beating Morton at home with the team we had out.
"I am disappointed that a cup competition has slipped past because we may not make it into Europe for the second half of the season and then we only have two competitions to play for rather than three.
"But I will get over it very quickly. We have too much to look forward to."