The Light Blues passed up a series of chances with Hoops goalkeeper Fraser Forster making key saves to keep the game scoreless at the interval.
Neil Lennon's 10 men held out for their 10th consecutive trophy win and his first success in the tournament as a manager, as Celtic delivered a psychological blow to Rangers, who have gone eight years without a major piece of silverware
This was another epic final between the two Glasgow giants.
The usual speculation and rumour surrounding team news before the game was ended when it was confirmed that Odsonne Edouard was only fit enough for the bench.
Few would have predicted the way Rangers bossed the first half as the rain refused to take a break.
The keeper almost let an angled-drive from Morelos slip through him and then Rangers defender Filip Helander headed a Tavernier corner past the post moments later.
Celtic were struggling to cope with Rangers' intensity.
Hoops defender Kristoffer Ajer headed a Christie corner over the bar but there was another Rangers chance in the 26th minute.
Morelos skipped past Ajer down the right but from an almost impossible angle he drove straight at Forster, with team-mates better placed.
Forster made another save from the Colombia international, this time from close range, as the aggressive Light Blues again got in behind the Celtic defence with ease.
Winger Mikey Johnston replaced the ineffective Elyounoussi for the start of the second half but Celtic soon had to defend with desperation.
Morelos came close when he returned a Tavernier cross back along the goal-line and then Forster again denied the forward with a terrific save from a close-range header.
In between that, a penalty shout when Morelos tangled with Celtic skipper Scott Brown inside the box was ignored by referee Collum.
There was no surprise when Edouard replaced Lewis Morgan in the 58th minute and his impact was immediate.
The French striker was chopped down by Goldson and Christie's free-kick from the left was steered in by Jullien, although there was a suggestion of offside.
Four minutes later, amid the bedlam, Morelos had the chance to break his Old Firm scoring duck when he was hauled to the ground by Frimpong, the 18-year-old Dutch defender sent packing before the Rangers striker saw his penalty parried away by Forster.
Celtic had the chance to double their lead in the 69th minute when Edouard set up Johnston on the break but with only keeper Allan McGregor to beat he lost his footing and shot wide.
The final moments were frantic and there was another Rangers penalty claim turned down when Johnston jumped with Morelos but Lennon's side survived to create another record and leave Rangers wondering what they have to do to end the Hoops' hegemony.