Arne Engels marked his first Celtic start with his first goal for the club
Celtic continued their 100% start to the season as the champions and Hearts experienced contrasting fortunes with VAR.
The visitors had a first-half penalty award overturned after Lawrence Shankland's header came off Liam Scales' arm.
And, after the break, Nicolas Kuhn's cross hit James Penrice's hand and the review resulted in record signing Arne Engels scoring his first Celtic goal from the spot.
Luke McCowan also opened his Celtic account with a late strike, confirming Hearts slipped a point adrift at the bottom of the Scottish Premiership.
Brendan Rodgers's side lead Aberdeen on goal difference and host Slovan Bratislava in Wednesday's Champions League opener.
An improved performance from the visitors, who take on St Mirren next, ultimately ended in a seventh straight defeat in all competitions.
Referee Colin Steven had been quick to point to the spot after the Scales incident but a prolonged VAR check ended with Celtic restarting on a dropped ball.
Neither Steven nor his assistant initially penalised Penrice, whose hand was outstretched, and it was more disappointment for Hearts as VAR ruled in the hosts' favour.
Engels, signed from Augsburg for a reported £11m, was given the responsibility rather than Reo Hatate and the new Belgium cap, 21, showed no signs of nerves as he rolled the ball past Craig Gordon.
Hearts' best chance came just before the McCowan goal from the edge of the penalty area. Musa Drammeh stretched out a foot to meet fellow substitute Blair Spittal's low cross but Kasper Schmeichel saved on his line.
And the clean sheet meant Celtic recorded five wins without conceding at the start of a season for the first time since 1906 - only the second time in their history.
Celtic take longer to get job done
Celtic, who had led by the 17th minute in all of their previous games this season, met greater resistance in Hearts than previous opponents. Still, the home side were profligate.
Kyogo Furuhashi fired wide after being sent clean through before Hatate's deflected shot was dealt with by veteran goalkeeper Gordon.
A quick throw-in by Alistair Johnston was flicked on by Kuhn for Engels to control and shoot but the Belgian's effort rattled Gordon's left-hand post.
Kyogo was denied superbly by Gordon and Kuhn's follow-up was cleared off the line by Penrice.
The converted penalty allowed Celtic the breathing space they sought and allowed Rodgers the chance to withdraw Kyogo, Hatate and Engels during the second half with Wednesday in mind.
McCowan was one of the beneficiaries and Gordon could only get a hand to his shot in off the post.
And Adam Idah thought he had netted his first of the campaign in added time when he met a Greg Taylor cross but an offside against Luis Palma earlier in the move kept the score at 2-0.
Grittier Hearts ultimately come up short
Hearts head coach Steven Naismith has been criticised this season for flitting between formations but, tactically, he made the right calls for the trip to Glasgow.
The 3-5-2 formation pitched Shankland and Kenneth Vargas, later replaced by Drammeh, as the attacking outlets and they battled hard to get their team up the park.
They had to make the most of set pieces and from one, Shankland's header was kept out by Schmeichel.
A long-range effort by Shankland effort had the keeper diving but it always looked like a comfortable save for the Dane.
But the Drammeh chance could have put a completely different complexion on the game and their start to the season.
A trip to Paisley next Saturday, while other clubs are on League Cup duty, at least gives Hearts the chance to make up some ground.
What they said
Celtic manager Brendan Rodgers: "By the rule, the officials were correct [to overturn the Hearts penalty award]. On the same token, if your arms up and the ball hits you, then it's going to be a penalty.
"[Reo Hatate] was the number one taker. That's the selflessness of Reo. The number two taker was Arne [Engels]. He stuck it away really well. A good performance by him.
"To be a Celtic supporter and get a goal at Celtic Park, what a special moment for [Luke McCowan] and his family."
Hearts head coach Steven Naismith: "The key moments in the game are the penalties. For me, they're both in the same camp. If one's given, they should be given. If one's not given, the other one shouldn't be. They both hit the hand.
"These moments are big in these games. The luck was against us."
