The Scottish champions took the lead through skipper Steven Davis before Ian Black levelled and Jamie Hamill grabbed the winner from the penalty spot.
Administrators Duff and Phelps are set to announce player redundancies early next week following several days of discussions with McCoist and PFA Scotland.
Rangers made two changes from the side who defeated Inverness last week for the clash against Hearts.
Maurice Edu was suspended and Kirk Broadfoot missed out with a knee injury, paving the way for Rhys McCabe to make his competitive debut, while Lee Wallace returned from injury to start against his former club.
Hearts were boosted by the return of Andy Webster from suspension, while Andrew Driver, Gary Glen and Adrian Mrowiec also started at Ibrox.
Rangers were boosted by another packed house after another difficult week off the park and they had the best of the chances in the opening spell.
Dorin Goian tried his luck early on, after the ball broke to him from a Sone Aluko corner, but his effort was tame and easily smothered by goalkeeper Jamie MacDonald.
McCabe then attempted to mark his senior team bow with a goal with a shot from distance that was also gathered by the goalkeeper.
After those early opportunities, chances were few and far between for both sides.
The only real incident of note was the first yellow card of the game issued by referee Crawford Allan to Darren Barr for a foul on Davis after 37 minutes.
But Rangers could have grabbed the opener just before the break when Aluko's pass found Salim Kerkar and he squared for Andy Little only for MacDonald to just manage to tip wide from the striker.
The breakthrough came on the half-time whistle when Davis met Little's cutback with a low strike into the back of the net.
Craig Beattie was handed his Hearts debut when he was thrown into the action in place of Barr for the second half.
Rangers then almost added to their lead when Carlos Bocanegra threw himself in front of McCabe's cross only to see his header flash wide.
The American defender came to the rescue at the other end when he nodded a Scott Robinson header off the line.
But the ball broke for Black to rifle into the back of the net from 20 yards to claim the equaliser after 58 minutes.
Rangers tried to haul themselves back into the game and Lee McCulloch claimed he was fouled in the box but there was no penalty award from the referee.
Allan did point to the spot at the other end when he ruled Goian fouled Webster and the Romanian was cautioned.
The match official then appeared to flash his yellow card at both Bocanegra and Goian for dissent but without showing a subsequent red card.
After the confusion, Hamill saw his initial attempt from the penalty spot blocked by Allan McGregor before firing home the rebound after 76 minutes to claim victory.