The midfielder's goal two minutes before the interval - after Jose Goncalves was fouled in the box - was the difference in a Boxing Day game bereft of clear-cut chances as Hearts put together successive victories in the league this term.
After a hard-fought first half, basement side Falkirk upped the pressure after the break but failed to find a way past a dogged Hearts rearguard.
Falkirk were first to threaten after three minutes when Vitor Lima pounced on Stewart's poor clearance and shot from 40 yards straight into the arms of Marian Kello.
In a tentative opening to the match, the visitors' first foray into the Bairns' half in the fifth minute ended with Ruben Palazuelos' cross being deflected wide for a corner.
However, David Templeton's set-piece was easily dealt with.
Hearts defender Marius Zaliukas then breathed a sigh of relief as Carl Finnigan pounced on his mistake to charge down the left flank but the striker's teasing low cross was picked up by Kello.
At the opposite end, Templeton's driven cross caused panic in the Falkirk defence before the danger was eventually cleared.
Hearts began to enjoy territorial advantage but they were aware of Falkirk on the counter-attack.
On a surge up the park in the 18th minute, Falkirk striker Carl Finnigan brought out a save from Kello from a tight angle.
And minutes later Ryan Flynn broke from a Templeton corner but his low effort was held by the goalkeeper.
After soaking up Hearts pressure, Falkirk upped the tempo to peg the Gorgie outfit back.
Firstly, Pedro Moutinho's free-kick from the edge of the area flew straight into Kello's arms.
From a corner kick, Ryan Flynn then shot wildly over the bar with Hearts relieved as they repelled another attack.
Hearts were given a great chance to break the deadlock when Goncalves was brought to the ground by Darren Barr inside the box.
Referee Callum Murray had no hesitation in pointing to the penalty spot and Stewart successfully converted his second penalty in a week, finding the corner of the net two minutes before the interval.
Falkirk were presented with a chance to restore parity just two minutes after the restart as Hearts striker Jamie Mole was penalised for hauling down Lima 25 yards from goal.
Toufik Zerrara struck a left-footed shot over the wall but Kello got down well to his effort.
Zerrara then picked out Finnigan, who had beaten the offside trap, but the marksman's delivery from the edge of the box was poor as Hearts mopped up the threat.
Falkirk were the livelier side following half-time as Hearts struggled to find any cohesion.
That did not dissuade manager Eddie May from making any changes, however, as Lee Bullen replaced Flynn.
Hearts manager Csaba Laszlo opted for an alteration of his own too, Suso Santana coming on for Templeton.
Santana was more direct than Templeton and his running won a free-kick 25 yards from goal.
Defender Marius Zaliukas had one thing on his mind as he approached the ball from a long run-up but his shot was deflected wide.
Falkirk goalkeeper Robert Olejnik was relieved to see Eggert Jonsson's header from Santana's corner drift wide as the midfielder got to the ball first.
Falkirk manager May made a tactical alteration as a result of Danijel Marceta's introduction as he pushed the towering Bullen into attack at the expense of Moutinho.
Marceta forced another corner for his side as his low shot was palmed behind by the outstretched Kello after the winger cut inside from 18 yards.
Hearts defender Ismael Bouzid should have done better with a free header from Santana's corner as his effort drifted off the top of his head and well wide.
The away team registered only their second shot on target in the dying minutes when Santana toe-poked an effort straight at Olejnik.