The home side had 27 corners and made umpteen chances but couldn't find a way past inspired goalkeeper Nicolas Caraux and were dumped by a Dougie Imrie penalty.
Morton had beaten only Cowdenbeath and Montrose all season and sit second bottom of the SPFL Championship, yet they embarrassed Celtic and ended their chances of winning the League Cup and a treble.
The tournament remains the only domestic honour Lennon has yet to win as a manager.
"I've got to be philosophical about it, it wasn't about their attitude or for the want of trying to win the game," said Lennon.
"We have to take our chances. We were flat-footed in the forward areas at times. That's something the new boys coming in have to adapt to very, very quickly."
That appeared to be a reference to Nir Biton, Tom Rogic and Dylan McGeouch, who all started. "I need to rely on the squad and I need to be able to give players a rest now and again. But if the ones coming in are going to perform like that then they are going to be sitting on the sidelines for quite a while.
"I'm not going to hammer the players. It has been a very demanding start to the season. It was a trophy up for grabs and you saw by the [strong] team I put out tonight that I wanted to win the game. That's the frustration for me but I will get over it very quickly. I have to. We have got too much to look forward to dwell on it. Congratulations to Morton. They defended brilliantly at times. But we should be beating Morton at home with the team we had out. We haven't and I will take criticism for that. That's fine." Imrie's penalty came after an Efe Ambrose handball but Lennon absolved him from blame and said he had slipped.
Morton manager Allan Moore revealed his chairman, Douglas Rae, had spoken to him on the eve of the match to reassure him that his job was safe despite their poor start to the season. "There were stories saying I was under pressure," said Moore. "He told me my job was 100% safe at the moment. I can understand Morton fans' frustrations. I don't need anyone to put pressure on me, I put pressure on myself. It's a work in progress and hopefully we get time to bring that through. To a man we were excellent tonight, especially after the league form we've had over the last couple of weeks.
"What a dressing room it was: that's why you come to play football. We got a bit of luck with the penalty. We lost a goal to [Queen of the South's] Iain Russell on Saturday who's probably smaller then me. Celtic were throwing guys in there at 6ft tall!"
Celtic supporters, meanwhile, could be allowed to stand during home games if the club successfully proceeds with a plan to create a safe-standing area. Peter Lawwell, the Celtic chief executive, revealed yesterday that the club could become the first in the SPFL Premiership to introduce safe-standing, almost two years after the Scottish Premier League relaxed its rules on all-seater stadiums. The SPL rules, absorbed this summer by the new SPFL, allow clubs to explore the sort of modern terracing areas common in the Bundesliga and elsewhere on mainland Europe. One option would be rail seats, which can be tipped up and locked for domestic fixtures but then converted back to seating for games in the Champions League or Europa League, where Uefa insist on all-seater grounds.
A safe-standing area could also allow Celtic to resolve some ongoing issues with The Green Brigade, who were briefly banned earlier this season because of issues around "lateral movement" along rows and aisles in their section of the stadium.
"We think that with some of the systems that are now deployed in Germany and other countries, it's now time to give them a try," said Lawwell. "We feel there is a new vibrancy in football that has come from Europe and is now in the UK.
These are new systems that are extremely safe and we are very keen to explore further implementing that at Celtic Park."
Celtic lifted their brief ban on
The Green Brigade after reaching an agreement over their behaviour in section 111 of the Parkhead ground, but Lawwell believes a safe-standing area could satisfy the desire among those and other supporters for a different way of watching games.
"They bring a lot to the game, to the event here," he said. "But there are certain aspects of the behaviour that we would deem not safe. This would be the ultimate solution."