A ninth goal in seven games for the Japanese international in what could well be his last match in a Celtic shirt.
Celtic's continued dominance was then reflected in the scoreline when Engels claimed his second goal in as many games. His shot from the edge of the area was firm and true and flew past a seemingly-unsighted Oxborough.
A triple substitution by Lennon at the break gave Dunfermline hope. Zak Rudden, Chris Kane and Shea Kearney all made an immediate difference before fellow sub, Alfons Amade, whistled a shot narrowly wide.
Iheanacho thought he had put the game beyond doubt only for his effort to be ruled out for an offside decision against Hyun-Jun Yang, who provided the assist.
The former Nigeria international didn't have to wait long for his moment of glory, though, finishing off a good Celtic move with some nifty footwork inside the box to claim Celtic's third and put the game well beyond a tiring Dunfermline.
After the nail-biting drama of the title win over Hearts last Saturday, this was a more comfortable afternoon for Celtic, but the victory will feel just as sweet. They have now won seven of the last 10 Scottish Cups.
If this turns out to be O'Neill's swansong, he has left the club with another two trophies to celebrate long into the summer.