The England goalkeeper broke Bobby Clark's previous record of 1,155 minutes without conceding a league goal before Leigh Griffiths scored his first Celtic goal to seal a 15th consecutive Scottish Premiership triumph.
The former Hibernian striker and boyhood fan celebrated emphatically after finishing impressively on a 57th-minute counter-attack.
That goal proved enough although Teemu Pukki doubled the lead with the last kick of the ball.
Forster had already broken former Aberdeen goalkeeper Clark's record, which he set in the 1970-71 season, after keeping Hearts out 31 minutes into his 13th successive league shutout.
The 25-year-old did not have a shot to save in that spell, although he fumbled a 21st-minute cross from Jamie Hamill before gathering at the second attempt.
Hearts also had some pressure around the half-hour mark but Callum Paterson could not find Paul McCallum with a low cross following a promising break and the record was assured when Sam Nicholson sliced the ball behind for a goal kick after a short corner.
More surprisingly, Hearts goalkeeper Jamie MacDonald, who conceded seven goals when Celtic last visited Tynecastle, was not seriously tested either during the first half.
Brad McKay twice foiled Griffiths with crucial interventions while Anthony Stokes had a shot deflected wide from 16 yards and Scott Brown shot over from just further out on the only other first-half occasions that Celtic threatened to get in behind.
Virgil van Dijk was just off target with a 25-yard free-kick and also a header following a corner from Kris Commons, who came closest for Celtic before the break when he hit a rasping long-range strike that dipped just over.
Celtic survived a penalty appeal three minutes before the half-time whistle following Van Dijk's clumsy tackle on Paterson, referee Kevin Clancy waving appeals away.
Both goalkeepers made their first saves of the game in the opening two minutes of the second half, Forster from Ryan Stevenson's long-range strike and MacDonald from a driven free-kick by Commons.
MacDonald was forced into a more testing stop when he parried Stokes' header following a deep free-kick by Commons.
And he was soon beaten after his side were caught out on the break not long after their own corner.
Stokes won the ball in the right-back position and beat a man before sending a diagonal ball for Griffiths to chase. The January signing from Wolves shrugged off Nicholson and took the ball round Danny Wilson before firing a powerful left-footed strike across MacDonald and into the corner from 16 yards.
Griffiths, who had been taunted by the Hearts fans throughout, slid in front of a section of home fans to celebrate before sprinting towards the dugouts with his thumbs raised at another section of the home support, before embracing manager Neil Lennon.
Lennon ordered his player to calm down but Griffiths soon followed three home players in the book for a trip on Nicholson and was then replaced by Pukki with the 5-1 scoreline of Hibernian's 2012 Scottish Cup final defeat by Hearts ringing in his ears.
Hearts forward Dale Carrick, on for the injured McCallum, caused some problems for Celtic but the hosts could not threaten Forster's ongoing record and MacDonald tipped over substitute Derk Boerrigter's strike as the visitors pushed for a second.
Forster survived a scare in injury-time when he missed his punch as Dylan McGowan attacked a corner but the ball drifted across the face of goal.
And Celtic went straight up the park to kill off the contest. Amido Balde robbed Hamill on the left wing and set up Pukki to slide the ball home.
Celtic will now push for more records as they seek to finish a top-flight campaign unbeaten for the first time since the 19th century and aim for their own record of 25 consecutive league wins, which they set in 2000-01.