Second-tier Falkirk deservedly led at the break after Ross MacIver and Finn Yeats twice gave them the lead either side of Paulo Bernardo's powerful strike for a much-changed Celtic team.
Kuhn was one of four substitutes to enter on the hour mark and he twice set up Adam Idah for a quickfire double to turn the game on its head before the German winger netted twice himself.
The scoreline was harsh on a Falkirk side who arguably posed Scottish Premiership leaders Celtic their toughest challenge of the season before the final 20 minutes.
"If there's any blame on the first hour, it's on me," Celtic manager Brendan Rodgers told the BBC. "I make the changes.
"It's a challenge for the guys coming in and I thought they gave everything. The players show a real hunger to work hard."
Celtic will play Aberdeen in the semi-finals while holders Rangers face Motherwell.
