The first Old Firm Scottish Cup final in 22 years offer very little goalmouth action in a tightly contested first half.
With the final goalless at the break, Rangers thought they had found the crucial breakthrough when Adballah Sima turned the ball in from a corner, only to see his effort ruled out following a VAR review.
Rangers carried the greater threat in the second period, but they were ultimately punished for failing to capitalise on their moments of promise.
Idah proved to be Celtic's hero, pouncing on a rebound to tuck the ball into the awaiting net and spark scenes of jubilation among those wearing green and white.
Celtic protected their narrow advantage in stoppage time to clinch a league and cup double, ensuring Rangers will end the campaign with just the Scottish League Cup to show for their efforts.
Sima denied by Raskin foul
When the two sides met at Celtic Park earlier this month, those in attendance witnessed three goals and a red card in a thrilling first half.
However, the first period failed to produce a similar level of drama as the two Old Firms rivals cancelled each other out in a cagey half at Hampden Park.
While chances were hard to come by, Joe Hart was still required to make a save from Fabio Silva's tame effort in the 18th minute.
At the opposite end of the pitch, Alistair Johnston had Celtic's best chance of the first period, forcing Jack Butland into a near-post stop with a deflected effort.
Rangers boss Philippe Clement opted to make a change at half time, swapping Cyriel Dessers for introducing Brighton & Hove Albion loanee Sima.
Clement must have thought he had pulled off a masterstroke when Sima guided the ball home from James Tavernier's deep corner just before the hour mark.
However, Sima's celebrations ultimately counted for nothing as the goal was chalked off or Nicolas Raskin's needless push on Hart.
Adah strikes late to clinch Scottish Cup
Despite being denied an opener, Rangers appeared to take confidence from that moment, and were arguably the stronger side in the second period at Hampden Park.
However, a resolute Celtic dealt with Rangers' moments of promise before carving out an all-important winner in the 90th minute.
After driving towards the edge of the box, Bernardo saw his fizzing effort parried by Butland, allowing Idah to pounce on the rebound and stroke home the winning goal.
The Scottish Premiership champions dealt with the remaining moments to claim the Scottish Cup for the second straight season.
As well as sealing a league and cup double, Celtic finished the campaign with an unbeaten record in the Old Firm derby, recording four wins and a draw in their five meetings with Rangers this term.