Celtic manager Brendan Rodgers felt Falkirk had posed their biggest challenge of the season but he took responsibility for a disjointed first hour after making eight changes.
Falkirk led at Celtic Park approaching the final 20 minutes but substitute Nicolas Kuhn produced two assists for Adam Idah and two goals himself as the Hoops set up a Premier Sports Cup semi-final against Aberdeen.
Strikes from Ross MacIver and Finn Yeats meant Falkirk scored as many goals in 45 minutes as Celtic’s previous seven opponents had managed in 630 minutes of action with only Hibernian and Slovan Bratislava finding the net.
John McGlynn’s William Hill Championship leaders deserved their lead and were comfortable until after Rodgers brought on Kuhn, Greg Taylor, Yang Hyun-jun and Arne Engels on the hour mark.
When asked if Falkirk proved their toughest test yet, Rodgers, whose side beat Rangers 3-0 earlier this month, said: “Yes, I thought Falkirk were excellent. I was expecting it, to be fair, which is why we tried to freshen the team up, because they give good energy and move the ball well, so they work you.
“It was a little bit disconnected for the first 60 minutes, and that’s on me, it’s not on the players.
“When you make eight changes like that… I wanted to get some players in and play in the stadium, get in their first minutes of the season, playing with the team.
“In some moments, we looked good, and in other moments, there was just that little bit of hesitancy. That comes from just not playing so much.
“But we knew we always had the bench to pull on if we needed it, especially in that last 30 minutes where Falkirk put so much into the game, and then the energy can maybe just start to get sucked a wee bit.
“We could then put some speed and tempo and rhythm into the game, and by the end, we were very good.
“It’s a huge credit to John, his players and his staff, because I had a real joy analysing them and seeing how they play, and seeing how they commit to attacking football.
“I really wish and hope that they can go on and get promoted, because they play a fantastic style.”
One of the players Rodgers brought in was debutant Alex Valle, the 20-year-old on-loan Barcelona left-back, who was outmuscled by Alfie Agyeman before both goals.
Rodgers admitted they were a “wee bit soft” in the lead-up to Yeats’ headed goal and that Valle will learn a valuable lesson.
He said: “It’s a crash course in British football, isn’t it? You might get (a free-kick) in Spain, but you certainly won’t get it in Britain. He’ll learn that, and he’s a good kid. He’s only young.
“He’s settled in really, really well into the city, into the club, and he’s there for us when he’s needed.”
Rodgers left both Cameron Carter-Vickers and Alistair Johnston out of his squad because of injury problems.
“Cam has been suffering a wee bit with his metatarsal, below his big toe,” he said, “He’s got a little bit sore, obviously playing back-to-back games so we’ll just assess that.
“Ali will be fine. He’s had a bit of an issue around his back, so he was having an injection but he’ll train Tuesday and Wednesday and be absolutely fine.”
McGlynn felt his team had done themselves and their fans proud.
“It speaks volumes for our team that we could come here and score two goals and Calvin Miller had a decent shot as well. We caused one or two problems.
“Bringing on the subs was a big thing in the game, four players coming on of that calibre. Our guys slightly get a little bit tired, which was only going to happen because obviously we worked so hard to stay in the game and to get in front in the game.
“And of course playing against a team like Celtic, a big park out there, their legs were probably going to end up coming off a little bit and I think that’s what happened.
“Celtic exposed us with the way they play and fair play to them but no criticism in any way from myself to these guys.”