Leigh Griffiths - with his first for Celtic - and Teemu Pukki secured the points as keeper Forster beat former Aberdeen stopper Bobby Clark's record.
Celtic had found clear-cut chances limited all afternoon but Virgil van Dijk's free-kick from 25 yards swerved narrowly past the post with Jamie MacDonald beaten after Jamie Hamill had whipped the legs from Kris Commons early on.
A few moments later Hearts defender Danny Wilson made a timely block to thwart Griffiths inside the penalty area before Scott Brown crashed the rebound over the crossbar.
Commons was next to try and make the breakthrough when he shaved the bar with his dipping drive shortly before Van Dijk headed an inch over the target at a Celtic corner.
Forster beat the 43-year record 31 minutes into the match and the big keeper was tested by a swerving 25 yard drive from Ryan Stevenson just after the break, which he did well to hold.
Forster moved past Clark's previous record of 1,155 minutes without conceding, which the former Aberdeen goalkeeper achieved in the 1970-71 season.
The England international had kept 12 consecutive Scottish Premiership clean sheets before the visit to Edinburgh, last shipping a league goal when Aberdeen's Niall McGinn scored past him at Celtic Park on November 23.
MacDonald then denied Celtic when he blocked a header from Tony Stokes on the line after the Irish striker drifted into space at the back post to meet a Commons free-kick.
The breakthrough came on 58 minutes when Griffiths used his strength to latch on to a long pass from Stokes pass before the former Hibernian favourite turned Wilson inside out and dispatched a powerful drive beyond keeper MacDonald.
Celtic substitute Dirk Boerrigter almost added a second on 86 minutes when his first touch in the game deflected off Dylan McGowan and clipped the bar after Adam Matthews had burst into the box to set up the chance.
Hearts came close to earning a point in stoppage-time when a long throw-in from McGowan dropped inside the six-yard boss beyond Forster, but Efe Ambrose came to the rescue to hook the ball clear with two Hearts players lurking.
Moments later Amido Balde disposed Hamill on the touchline and strode towards the penalty area before squaring a pass to Pukki, who kept his composure to tuck the ball into the corner of the net.