The game turned in the 28th minute when, seconds after Gorgie midfielder Suso Santana had hit the post with a drive, James Forrest opened the scoring by knocking in a Daryl Murphy cross from close range.
Hearts boss Jim Jefferies was sent to the stand by referee Willie Collum immediately after for complaining about an offside in the lead-up to the goal but Shaun Maloney grabbed a clinching second before the break and in the dying stages Paddy McCourt, on for Anthony Stokes, scored a wonderful third.
Stokes was making his Celtic debut - the 22-year-old Irishman, signed from Hibernian just before the transfer window closed, replaced Georgios Samaras to line up in attack alongside former Sunderland team-mate Murphy.
Tynecastle boss Jefferies was without left-back Lee Wallace, who suffered a cruciate ligament injury playing for Scotland against Liechtenstein at Hampden in midweek.
Ruben Palazuelos replaced Craig Thomson, who started on the bench.
Before kick-off there was an enthusiastic minute's applause to commemorate the 25th anniversary of the death of legendary Celtic manager Jock Stein.
In the 10th minute, Stokes finished off a passage of play by driving just wide of the post from distance.
But the visitors responded and in the 18th minute, Kevin Kyle's header from a David Templeton corner was well saved at the back post by Celtic keeper Fraser Forster, who moments later had to tip Ryan Stevenson's attempted lob over the bar.
In the 27th minute Suso Santana was set up 25 yards from goal by Stevenson and although his low drive beat Forster, it rebounded off the keeper's right-hand post - and then Celtic raced up the park to grab the opener.
Maloney sent Murphy running down the right and into the penalty area and, when his ball came across the six-yard box, Forrest was on hand at the back post to bundle the ball into the net.
Hearts boss Jefferies was then sent to the stand for complaining that there should have been an offside in the lead-up to the goal.
But there was no controversy around Celtic's second goal a minute from the break when Maloney was sent clear by Stokes, the Scotland international rounding Marian Kello before slotting into the net.
It looked a long way back for the Edinburgh side and they should have been 3-0 down in the 50th minute when a clever Stokes flick released Murphy, but Kello guessed correctly as the Irishman drove into the box and saved at his feet.
Then, in the 57th minute, Maloney had at least two chances to grab his second goal following a Forrest cross from the right into a packed Hearts box, but Kello pounced on the loose ball.
Moments later Hearts had a claim for a penalty when Celtic defender Emilio Izaguirre clashed with Templeton at the edge of the box but it was declined by referee Collum.
In the 71st minute Celtic substitute Samaras, on for Murphy, shot just over the bar and then had the ball in the net minutes later only to be ruled offside.
The home side continually threatened to add to the lead and Kello made a fantastic save from Samaras after Maloney had set him up eight yards out.
The Greece striker was again foiled by the Hearts keeper a minute from time, but Celtic substitute Paddy McCourt, on for Stokes, scored a wonderful solo goal right at the end to make it 3-0.