Brendan Rodgers claimed St Mirren were the best team Celtic have faced in Scotland this season following their Scottish Cup win over the Buddies at Parkhead.
Harry Davis stunned the Hoops support with his goal from close range in the 13th minute, and only the crossbar prevented the Championship's bottom side doubling that lead in the 53rd minute.
However, two goals in barely a minute through Mikael Lustig and Scott Sinclair ended any prospects of a shock defeat for the runaway Premiership leaders, with Moussa Dembele and substitute Leigh Griffiths adding further goals to take them into the semi-final.
Celtic extended their unbeaten domestic run since the start of the season to 34 games and are 27 points clear of second-placed Aberdeen as they begin preparing for the visit of Old Firm rivals Rangers next week.
Rodgers praised former Alloa boss Jack Ross for the work he has done with Paisley side St Mirren since taking over last October.
He said: "They are the best team that we have played domestically, including Premiership teams, without any shadow of a doubt.
"How they are at the bottom of the league I do not know. They are a very good side.
"Jack is obviously a very good coach. He's gone in there and he's got them very well organised.
"There were a couple of moments in the game that made me think of the level of coach. It's just that extra level of detail which can define coaches. I saw that in his team today.
"They have good speed, they have young players with enthusiasm and energy that can hurt you.
"They have the boys Stephen McGinn and Stephen Mallan in midfield who are technically gifted. They are not worried about hooking it on or smashing it. They play football.
"They have been our biggest threat domestically this season for sure.
"Let's be clear, we ended the game very well in the second half. We were 4-1 up and it could have finished 6-1 or more.
"I want to compliment my players first as they were outstanding."
Rodgers confirmed that goalkeeper Craig Gordon was set to sign a new deal next week.
He said: "I am led to believe that everything is agreed and done, it is just a case of the formalities, the paperwork and that is it."
Ross admitted Mallan's free-kick coming off Dembele and smacking the bar when his side were leading 1-0 was a crucial point in the game.
He said: "You just don't know how things would have panned out after that. I still think we would have had to withstand a lot of pressure because at that stage Leigh had been introduced and that changed the dynamics of the game.
"However, I spoke beforehand of what you needed to do to win here: play very well and have good fortune.
"That was perhaps that little extra fortune that we needed on the day.
"But I am just pleased that the two goals at the end were disallowed because I think it would have put an unfair reflection on things.
"I thought four was even slightly harsh on us."