United took the lead through substitute Andis Shala but Rangers appeared to be cruising towards the last four when Kris Boyd grabbed a double from the penalty spot and Nacho Novo netted a third for the holders.
But the Tangerines - who were battered 7-1 in their last visit to Ibrox - were thrown a lifeline thanks to a Steven Whittaker own-goal, before Kovacevic levelled late on.
Rangers headed into the tie without Madjid Bougherra after he failed to recover from injury in time, while Novo replaced DaMarcus Beasley.
United were without Andy Webster - who was ineligible under the terms of his loan deal - and the injured Paul Dixon, while David Goodwillie was absent from the squad. Sean Dillon, Darren Dods and Jon Daly were added to the side.
The visitors were forced to make a change almost immediately when Damien Casalinuovo appeared to pull up with just 56 seconds gone and was replaced by Shala with three minutes on the clock.
United goalkeeper Dusan Pernis was called into action to claw a Steven Davis effort away from goal before the ball broke for Whittaker but he failed to add to his 11 goals this season by rifling just over the crossbar.
Rangers continued to pile on the pressure and Kenny Miller whipped an inviting ball into the box for Novo only for the Spaniard to glance his header across goal and narrowly past the upright. Miller then went on a great run only for his weak shot to trickle wide.
The Ibrox side were left rueing those missed opportunities when United surged into the lead after 24 minutes.
Whittaker's slack pass-back had been intended for Danny Wilson but was intercepted by Daly, and he slipped the ball into the path of Shala who fired low under the body of goalkeeper Allan McGregor.
The home side were back on level terms with 34 minutes gone when Miller was fouled by Pernis and referee Dougie McDonald pointed to the penalty spot, before booking the goalkeeper.
Boyd stepped up for the spot-kick and his powerful, low strike left Pernis with no chance as he ended his mini goal drought.
The striker claimed his second of the day - another penalty - two minutes before the break when he dusted himself off to convert after being bundled over by Dillon.
United tried to haul themselves back into the tie after the interval and were not too far away when an opportune strike from Craig Conway bounced off the crossbar on the way over.
But hopes of a comeback were dealt a blow when Rangers claimed their third of the day just three minutes after the restart.
Novo collected a nice pass from Davis before dispatching an angled drive into the bottom corner, with Garry Kenneth's attempts to race back and hook to safety proving to be in vain as he ended up in the back of the net along with the ball.
But United refused to give up and pulled a goal back after 63 minutes when a ferocious volley from Morgaro Gomis took a deflection off Whittaker and whistled past his own goalkeeper.
Rangers were stunned when United drew level with 10 minutes to go. Conway's corner found Francisco Sandaza and his header crashed off the crossbar before Kovacevic nodded home the rebound.
Both teams will do it all again at Tannadice on Wednesday March 24 to determine who will progress to the last four of the tournament, with the holders having required two attempts to overcome both Hamilton and St Mirren in previous rounds.