Bjorn Johnsen headed the hosts into a 19th-minute lead, but Partick were much the better side in the second half, equalising through Sean Welsh. The Jags may well have left the Capital with all three points had it not been for two fine saves by Hearts keeper Jack Hamilton.
Hearts made two changes from the side that lost 2-0 to Rangers at Ibrox seven days ago, with Liam Smith coming in for Faycal Rherras and Prince Buaben replacing Robbie Muirhead in a 4-4-1-1 system. Partick made three alterations to their line-up, with Ade Azeez, Ryan Edwards and Danny Devine coming in for Ziggy Gordon, Chris Erskine and David Amoo.
The hosts had the first tangible chance on 15 minutes when Callum Paterson glanced a Don Cowie cross just wide of Tomas Cerny's post. However, it wasn't long until they opened the scoring on 19 minutes. Paterson released Liam Smith down the right-hand side and he managed to ride Callum Booth's sliding tackle before sending a whipped cross that Johnsen steered home with his head into Tomas Cerny's right corner of the net.
Hearts could've extended their lead right on half-time when Don Cowie intelligently found Arnaud Djoum free in the penalty box with a clipped cross, but the Cameroonian's shot was turned past the post by Cerny. From Cowie's resultant corner, Igor Rossi rose highest, but his looping header struck the bar and Partick scrambled the ball away.
Hearts made a change at the interval, with Conor Sammon replacing Buaben. Within three minutes, however, Partick - who pushed defender Adam Barton into midfield as they adapted their formation - had equalised. Ryan Edwards swung in a cross from the left and Welsh stooped lowest to guide the ball past Jack Hamilton and into the net.
Partick could have increased their lead a few minutes later when Kris Doolan released Azeez, but Hamilton stuck out a leg and diverted the striker's net bound shot wide.
The visitors were buoyed by their equaliser and they exerted pressure on a clearly nervy home side. Doolan and Azeez nearly combined again, only for John Souttar to put in enough of a challenge to put the former AFC Wimbledon man off his stride and the chance came to nothing. Then Edwards fired a rasping drive inches wide as the Tynecastle crowd voiced their displeasure.
The boos rang louder on 74 minutes as Cathro swapped Johnsen for Muirhead. Tony Watt was also introduced for Jamie Walker as the Jambos tried to regain the initiative. Muirhead nearly created a goal for Hearts three minutes later, slipping in Smith on the left. He jinked inside but his shot was a foot too high as it flew over Cerny's bar. Djoum then stung the palms of the Czech keeper with a low drive.
Partick still looked the more threatening going into the closing stages and came so close to scoring three times. First Booth rattled the underside of bar with shot from just outside the box and as Hearts failed to clear their lines, Hamilton made a tremendous save, flying low to his right turn Erskine's effort wide. From the corner, an unmarked Liam Lindsay headed wide from eight yards out.
Hamilton was Hearts' saviour yet again on 86 minutes. Edwards' slide-rule pass sent Doolan clean through on the keeper, but the Scotland squad member made himself big and blocked, with Booth unable to steer the bobbling rebound home.
Hearts had one final chance before the end, but Perry Kitchen's net-bound effort was blocked, leaving the Gorgie side in fourth place, seven points behind Rangers in second spot.
Hearts (4-4-1-1): Hamilton; Paterson, Souttar, Rossi, L Smith; Cowie, Buaben (Sammon 46), Kitchen, Walker (Watt 74); Djoum; Johnsen (Muirhead 74). Subs: Noring, Rherras, Ozturk, Nowak.
Partick (3-5-2): Cerny; Devine, Barton, Lindsay; Elliot, Welsh, Osman, Edwards, Booth; Azeez (Erskine 67), Doolan. Subs: Stuckmann, Gordon, McDaid, Wilson, Lawless, Amoo.
Referee: D Robertson.