Celtic manager Neil Lennon told his players he was "really proud" of them after their extra-time victory over Dunfermline.
The Fifers were six minutes away from taking the Betfred Cup holders to a penalty shoot-out before James Forrest's deflected strike earned Celtic a 2-1 second-round victory.
It was the first time Celtic have been taken to extra-time during a 28-match winning run in domestic cup competitions spanning three years, and Lennon was happy to see his players pass a stiff test of their character on the back of their Champions League exit.
The Hoops were again slow to get going but Mikey Johnston settled nerves with a brilliant curling strike 10 minutes into the second half.
However, Dunfermline equalised through Tom Beadling in the 77th minute with their only shot on target and frustrated their hosts until the 114th minute.
Lennon, whose side hit the target with 11 of their 36 shots, said: "I'm delighted. We dominated the game. I think Dunfermline scored with their one attempt on goal and that happens in cup football.
"I am really proud of the players because they had to show a lot of character.
"I know some of them weren't at their best but that's totally understandable. I have been in that situation myself after a Champions League game, you can be a bit leggy and a bit fatigued.
"We didn't deserve the game to go to extra-time and then we showed a lot of character to go on and win the game.
"You have to give Dunfermline a lot of credit, they defended brilliantly and came with a game plan. We didn't cut them open as much as we'd like but when you have a block of 10 in front of you..."
Lennon assembled his players on the pitch after the final whistle and expressed his pride in their work.
"They have had a real test of character and come out the other side," he said.
"They won't see it the way I see it. I played in a great Celtic side and we had days like that. I remember a penalty shoot-out against Partick Thistle and it went to 10-8 or something and they were a Championship side. I got knocked out by Morton in my last spell here.
"They have come through a very, very stiff examination, psychologically rather than a football one."
Dunfermline's equaliser came just after Kristoffer Ajer went off with his arm in a makeshift sling, but Lennon is hopeful the Norwegian will be fit for Thursday's Europa League visit of AIK.
"He popped his shoulder," Lennon said. "I think it's back in place now so we are hoping he should be OK for Thursday. He's a really important player for us, obviously."
Dunfermline manager Stevie Crawford was also proud of his team.
"We came with a game plan and we knew we were going to give up chances to try and restrict Celtic getting the ball in wide areas," he said.
"We did that very well and restricted them to chances outside the box.
"We got back in the game and went two up front and we have given Celtic a fright."