Rangers were quick out of the blocks, Morelos getting in behind from a long ball forward, but he couldn't get much of a contact on his finish and Joe Lewis made a comfortable save after just 20 seconds.
The Dons responded in the third minute, Ryan Hedges looping the ball forward into the centre circle. Curtis Main produced a clever pass from there to set Cosgrove off on the run, but Goldson was able to force him wider than he wanted to go and from 20 yards out, his shot was always arcing away from goal.
Hagi was next to try his luck four minutes later, unleashing a rasping drive from 30 yards out after the Dons lost possession in midfield, the ball flying over the bar as the game continued in enterprising fashion. Aberdeen were able to put the visitors under some pressure from there, neat interplay between Hedges and Wright causing Rangers concern, but the Dons couldn't quite manufacture a shooting chance.
Wright was causing more trouble in the 11th minute, getting past Goldson down the left and winning a corner with a dangerous cross. Lewis Ferguson took the flag kick, Tommie Hoban glancing his header wide under pressure at the back post. The Dons were pressing again a minute later, Cosgrove driving into space but his pass was a fraction too heavy and Wright couldn't quite catch up with it.
Rangers almost grabbed the lead in the 23rd minute, Kent wriggling free to get a shot away from 12 yards, Jonny Hayes there to clear it brilliantly off the line. Tavernier came in with the follow up shot, Lewis down smartly to his right to scoop the ball away to safety.
A clever one-two between Morelos and Aribo got Morelos into the box in the 26th minute. Ryan Hedges laid the slightest touch on the striker, which looked accidental, but Morelos went down and referee Beaton pointed to the spot. The penalty award was expected, but the straight red card which also followed for Hedges was incredibly harsh.
Tavernier stepped up, sent Lewis the wrong way, but put the ball well wide of goal, a let off for the Dons.
But Morelos did put the league leaders in front after 32 minutes. Tavernier drifted in from the right to pick out Kent in the centre of the box. He fed Morelos who, from around the penalty spot, struck a first time shot on the turn into the bottom corner.
Tavernier was the first Rangers man in the book after 39 minutes when he clipped Hayes as the Dons were looking to counter-attack. Main followed him into the book a minute later as he challenged with Barisic for the ball. That was Main's last act of the game, replaced by Shay Logan in the 44th minute as the Dons went to a back four.
HALF-TIME: ABERDEEN 0 RANGERS 1
Five minutes into the second period, there was a mountain for Aberdeen to climb as Rangers made it 2-0. Aribo speared a low ball through the box, Kent with a clever backheel to send it on its way through to Morelos. His first touch was immaculate and from there, he again clubbed the ball across goal and into the bottom corner.
Kent was having a strong afternoon for the visitors and on 55 minutes, he ran across the top of the box before firing off a left footed drive at goal, Lewis making a brilliant save to his right to push the ball behind. Aberdeen weren't far away from pulling a goal back two minutes later. Cosgrove won a free-kick after a run deep into Rangers territory and the Dons took it quickly, Cosgrove nudging the ball to Hayes whose ferocious drive across goal worked McGregor who had to go full stretch to push the ball away. Moments later, Kennedy replaced Wright for the home team.
Rangers were enjoying the lion's share of possession, making the extra man count, but the Dons worked hard to retain a good defensive shape to keep them at arm's length and to retain a sliver of a chance in the game. Hagi went looking for a second penalty with an outrageous dive in the 66th minute. Referee Beaton was quite right not to give it, but where was the punishment for simulation?
No matter, Aberdeen were back in the fight a minute later with a well crafted goal. A great ball forward from Logan found Cosgrove down the right. He moved the ball inside to Hayes, darting into space and, having drawn McGregor, he rolled the ball into the path of Kennedy who finished clinically from the edge of the box to reignite the game.
The Dons were beginning to ask a few more questions of a nervy looking Rangers now, a tactically intelligent and accomplished display from the ten men hosts keeping them very much in the match. Ferguson collected a booking in the 81st minute as Kent cleverly ran across him to win a free-kick around halfway, then Rangers withdrew Hagi to bring on Zungu to shore things up.
Rangers finally got themselves going as a threat again in the 87th minute, Kamara surging into the box and forcing a good low save from Lewis at his near post. Edmondson was then introduced for the last couple of minutes in place of Ash Taylor as the Dons looked for a last charge for the equaliser, but with surprisingly only two minutes of added time, there was no opportunity to fashion just one last chance.