First-half goals from Stuart Findlay and Lawrence Shankland saw the Edinburgh side take control before Youssef Chermiti got one back in the final seconds of the game.
There were three changes from Monday’s victory over Hibernian, with James Tavernier, Nico Raskin and Bojan Miovski coming into the line-up.
It was a welcome return for the striker, who had netted three goals in his last two matches before picking up an injury against Ferencvaros in the UEFA Europa League.
That meant Rangers started with Jack Butland in goal behind a defence of Max Aarons, James Tavernier, Dujon Sterling, Emmanuel Fernandez and Jayden Meghoma.
Nico Raskin joined Connor Barron and Mohamed Diomande in midfield, while Mikey Moore was in attack alongside Miovski.
It was almost the perfect start for Rangers in the capital, as a corner saw the ball end up in the back of the Hearts net. The set piece was whipped into the area by James Tavernier, with Emmanuel Fernandez glancing a header towards goal, which Bojan Miovski touched in at the back post. Unfortunately for Gers, the striker had strayed just offside and after a VAR check the goal was disallowed.
At the other end, the home side had their first opportunity as Alexandros Kyziridis cut inside from the right to curl a strike towards goal but it was wide of Jack Butland’s post.
Another set-piece opened up a chance for Rangers, as Tavernier sent an inviting ball into the box from a free-kick, but Raskin was unable to keep his header on target.
It had been a positive spell forRöhl’s men, with Mikey Moore seeing a shot pushed away by Alexander Schwolow before Meghoma’s rebound effort was diverted out for a corner.
However, with 38 minutes played, it was Hearts who took the lead. A short corner was played between Alexandros Kyzirdis and Claudio Braga, with the former delivering the ball into the box, and Stuart Findlay was able to get a strong head to the cross to send it beyond Jack Butland.
It was a disastrous few minutes for Rangers, as Hearts doubled their advantage four minutes later. A free-kick from the half-way line was sent deep into the box and Claudio Braga got a head to it to loop it over the defense and into the path of Shankland, with the Hearts captain making no mistake in finishing from a tight angle.
Röhl turned to his bench at half-time as Mohamed Diomande was replaced by Youssef Chermiti, with captain James Tavernier slotting into midfield.
Connor Barron sent a shot wide as a corner broke to the midfielder outside the box, before Shankland passed up a glorious opportunity to increase the league leaders’ advantage as he had space in the box to shoot but sent it just wide of Butland’s left-hand post.
There was another change for Rangers with 30 minutes remaining as Djeidi Gassama replaced James Tavernier.
Butland pulled off a good save as Hearts broke forward in numbers from a Rangers corner, with Shankland’s right-footed strike pushed away by the goalkeeper.
Danilo and Thelo Aasgaard were next to be introduced for Rangers as they continued to search for a route back into the game.
Emmanuel Fernandez again had an opportunity from a corner but didn’t get the connection he would’ve hoped for as it went wide after a good delivery from Thelo Aasgaard.
A final change saw Dujon Sterling, who is still continuing his recovery from a seven-month injury layoff, replaced by Clinton Nsiala.
With just seconds remaining at Tynecastle, Youssef Chermiti found the back of the net with a low finish, but it was too late for Rangers to rescue anything from the match.
Next up for Rangers is Motherwell at Ibrox on December 27.
RANGERS: Butland, Aarons, Tavernier (Gassama, 63), Sterling (Nsiala, 81), Fernandez, Meghoma, Raskin, Barron (Aasgaard, 74), Diomande (Chermiti, 46), Moore, Miovski (Danilo, 74).
Subs: Kelly, Adamson, Dowell, Curtis
Heart of Midlothian: Schwolow, Kent, Kingsley (McCart, 87), Shankland (Kabangu, 90), Braga (Forrest, 81), Devlin, Steinwender, Findlay, Magnusson (Baningime, 81), McEntee, Kyzirdis (Kerjota, 87).
Subs: Gordon, Borchgrevink, Spittal, Wilson.