The Edinburgh side headed into Saturday's match at Tynecastle boosted by last weekend's win over Celtic, which allowed them to book their place in the Scottish Cup final against city rivals Hibernian.
A Sone Aluko goal and an Andy Little double sealed the win for administration-hit Rangers, while a penalty miss from cup hero Craig Beattie only added to Hearts' despair.
Rangers made three changes from the side which defeated St Mirren in their last outing a fortnight ago for the clash with the Jambos.
Steven Whittaker and Lee Wallace both returned from injury to start the match and Rhys McCabe was drafted into the side in place of injured skipper Steven Davis.
Hearts made one change from last week, with match-winner Beattie given the nod ahead of Scott Robinson.
Rangers could have had a penalty in the opening five minutes when Darren Barr appeared to handle a Maurice Edu header in the box but referee Calum Murray did not point to the spot.
The visitors continued to push forward and Little squared into the path of the on-rushing Lee McCulloch in front of goal but he connected awkwardly and goalkeeper Jamie MacDonald was able to gather.
Kyle Bartley then saw his volley cannon off former Ibrox defender Andy Webster on the line, before McCulloch's shot flashed across goal and wide.
Rangers surged into the lead after 28 minutes when they were handed a free-kick just outside the box for a foul by Webster, which was hotly disputed by the Hearts player.
Aluko stepped up for the set-piece and sent a superb curling effort past the grasp of MacDonald to leave the net bulging.
Rangers could have doubled their lead when Whittaker supplied a cut-back for McCulloch and, with only the goalkeeper to beat, he looked on in disbelief as MacDonald just managed to tip the shot wide.
The honours went to Little instead when he collected a Wallace cross at the near post and scooped a left-footed effort past MacDonald and into the net with 35 minutes gone.
Hearts made their first change at the break when Stephen Elliott was withdrawn for the introduction of Suso in the second half.
The home side had failed to create much in the way of goal-scoring opportunities in the first half but Beattie was not too far away shortly after the re-start with a shot that fell just wide of Allan McGregor's left-hand post.
The former Celtic striker had another chance when he picked up Ian Black's pass only to blast well over the crossbar from 15 yards, failing to trouble the Rangers goalkeeper once again.
Hearts were determined to haul themselves back into the game and Rudi Skacel forced a decent save from a full-stretch McGregor with a powerful long-range drive.
The Jambos should have pulled a goal back when the referee pointed to the penalty spot for an Aluko trip on substitute David Templeton just inside the area after 75 minutes.
But Beattie, who had been the hero with the winner from the penalty spot against Celtic last weekend, saw his spot-kick crash off the crossbar on this occasion.
There was no way back for Hearts when Wallace found Little in plenty of space in front of goal and all that was required from the Northern Irishman was to slot into an empty net to put the gloss on the win with two minutes to go.
