Adam Matthews fired home a wonderful strike from 20 yards in the 19th minute but Celtic subsequently failed to take control at Gayfield Park.
And they were grateful for a 66th-minute decision by referee Craig Charleston, who ruled out Steven Doris' close-range finish for an earlier infringement on Celtic goalkeeper Fraser Forster.
Forster himself averted extra-time when he saved a one-on-one with Connor Birse as Celtic edged through to set up a trip to Raith Rovers.
Lennon cut a frustrated figure on the touchline as a number of Celtic moves broke down, but he felt the freezing conditions played a huge part in their loose passing.
"The longer the game went on, the more difficult the pitch became. It was almost like an ice rink," Lennon said.
"You could see a lot of the passes just running away from people because of the pace the frost was bringing to the grass. So it became a real leveller. It was a real test of their character and they came through it very well."
And Lennon felt Charleston was correct in the major decision. He added: "I thought it was a free-kick and we relied on Fraser to make a good save with his legs. But we had good chances.
"We had a goal disallowed (from Lassad Nouioui) and I would need to see again if it was offside. We had two or three chances to put the game beyond all doubt, but the ball just kept running away from people.
"It just became a difficult night for us, but we are in all the competitions going into the new year and I can't ask for any more than that."