A scrappy game was decided by an error as Jack Butland fumbled Paulo Bernardo's effort and Idah pounced for his ninth goal of a loan spell from Norwich.
"I said I'd come here to try and score as many goals as I could, I didn't think I'd score a goal like this in front of all these fans. It's an amazing feeling," said Idah, who only made the move to Glasgow in January.
"It's hard to put into words. Anyone here can see what it means to the fans, to the players. I've always supported Celtic, it's amazing!"
Rangers thought they had taken the lead earlier in the second half when Abdallah Sima turned home a corner at the far post.
But after a VAR review, Celtic goalkeeper Joe Hart was handed a reprieve in the final match of his career as the former England international had been pushed by Nico Raskin.
Celtic were far from their best but maintained their unbeaten record in five meetings against Rangers this season, which has also secured them direct qualification for next season's Champions League by winning the Scottish Premiership.
Victory also continued Brendan Rodgers' remarkable Scottish Cup record across two spells in charge of the Hoops.
Rodgers has never lost a Scottish Cup tie in 17 matches and takes his trophy haul as a Celtic boss to nine in three-and-a-half seasons.
Celtic's win also has historical significance as they move level with Rangers' total of 118 trophies.
The head-to-head between the clubs is also now level after 441 meetings with 169 wins each and 103 draws.
