With just one defeat in nine games, Aberdeen took impressive form into this powderkeg fixture and knew a win would have returned them to third place.
Rangers, with one eye on the second leg of their Europa League quarter-final at Athletic Bilbao on Thursday, made nine changes and were skippered by former Dons midfielder Connor Barron.
The unfamiliar visiting side survived early let-offs when Shayden Morris fired narrowly wide after a great turn then Mats Knoester volleyed over from a few yards out.
But Aberdeen's pressure told when Clarkson curled a wonderful free-kick into the top corner from 20 yards.
And after McCausland's dismissal, Gueye tapped in from close range just before the break to double the home side's advantage.
Igamane had the ball in the net for Rangers in first-half added time, but it didn't count as Findlay Curtis was judged to have run the ball out of play in the build-up.
The Morocco striker wasn't to be denied, though, as he outmuscled the Aberdeen defence to reduce the deficit early in the second half.
That sparked a few jitters among the home support at a packed Pittodrie as, despite being a man down, Rangers started to make their mark in the game.
Curtis fluffed an effort from a good position while at the other end, Keskinen, Kevin Nisbet and Alexander Jensen sent shots wide.
Chances continued to come and Aberdeen thought they had restored their two-goal advantage when Nisbet smashed the ball into the net, but his goal was chalked off for a handball by the Millwall loanee.
And that was to prove crucial as Hagi levelled in the sixth and final minute of added time, following up with a fine finish after Cyriel Dessers had struck the post with a low drive, as Celtic's title champagne was put on ice.