After Old Firm rivals Celtic thumped Ross County 6-0 to regain top spot on Saturday, Steven Gerrard's Light Blues went into Sunday's lunchtime kick-off knowing a victory would take them back to the summit.
But they found themselves on the back foot early on as Hearts flew out of the blocks in a breathless first half, hitting the crossbar inside the first two minutes before opening the scoring four minutes later through Ryotaro Meshino.
And despite creating a flurry of second-half chances, Rangers had to settle for a point which takes them level on 22 points with leaders Celtic.
Hearts made a lightning-quick start to the game, signalling their attacking intent in the opening minute when Jake Mulraney burst into the box on the left side and won a corner.
After a frantic opening 15 minutes, the game settled down towards the middle of the first half and Rangers were limited to half-chances as they went in search of an equaliser.
The visitors almost hauled themselves level in the 34th minute when Steven Davis' diving header from outside the box spun towards goal but it nestled into the side-netting.
Rangers continued to threaten and eventually got their reward in the 39th minute when September's Ladbrokes Premiership player of the month Morelos coolly slotted in at the back post after Borna Barisic's corner had been knocked down into his path by Nikola Katic.
Rangers asked plenty of questions of Hearts' defence early in the second half, with Davis and Morelos both seeing efforts deflected behind before James Tavernier's 61st-minute free-kick was blocked in the box.
There were huge cheers from the 3,045-strong away following in the 62nd minute when winger Ryan Kent stepped off the bench to make his return from a hamstring injury.
And the summer signing from Liverpool created his first opportunity four minutes later when he beat two Hearts defenders before rifling a fierce shot wide from the left flank.
Rangers felt they should have had a penalty when Scott Arfield's path was blocked in the box following good build-up play between the Canada international, Kent and Morelos.
Hearts goalkeeper Pereira made a smart save to keep out Connor Goldson's header in the 80th minute as Rangers stepped up their search for a winner.
A quick counter-attack from Hearts petered out as Rangers were able to clear in their box after Ikpeazu had been played in by substitute Andy Irving.
The home side could have won it late on when Christophe Berra nodded towards goal in added time but it was straight at McGregor.