Early goals in both halves eased Celtic into the Scottish Cup final, where they will face Dundee United.
Chris Sutton (right) opened the scoring for Celtic
Chris Sutton made a significant return to the holders' side, heading home following poor marking by Hearts at a corner after just three minutes.
Goalkeeper Craig Gordon let Craig Bellamy's shot on the turn slip in at his near post just after the break.
Deividas Cesnauskis replied with a superb chip, but Celtic held on to avenge last week's league defeat.
Sutton had come into the Celtic starting line-up after recovering from injury, replacing Aiden McGeady in midfield.
Joos Valgaeren was preferred in defence to Stephane Henchoz, the former Liverpool defender having been taken off early in last weekend's 2-0 league defeat.
Hearts head coach John Robertson brought back Paul Hartley in place of Marius Kizys, the Scotland midfielder having been suspended for the victory at Celtic Park.
Hearts fans ensured a stormy opening by jeering during the minute's silence in memory of the Pope.
And they were quickly silenced after kick off.
Alan Thompson tapped a corner to Stilian Petrov and slack marking allowed Sutton to slip between Steven Pressley and Andy Webster - and in front of goalkeeper Craig Gordon - to power his header home from six yards.
Hartley sent Mark Burchill clear down the wing and the former Celtic striker's cross was touched on by the midfielder, but Joe Hamill's strike from the edge of the box struck Valgaeren.
There were few chances in a scrappy and lacklustre first half.
But Hartson was inches away from turning the ball into the net from close range after a Neil Lennon pass during a penalty-box scramble.
Celtic looked to have killed off the game four minutes after the break.
Bellamy turned sharply in the box and his shot took a slight deflection off Andy Webster and slipped through the arms of Hearts' young Scotland goalkeeper.
A Bellamy cross struck Hearts' crossbar, Stanislav Varga had a header cleared off the line by Lee Wallace and Alan Thompson forced Gordon into a close-range save as Celtic threatened to run away with the game.
But the introduction of Lithuanian pair Cesnauskis and Kizys gave Hearts new impetus.
Jamie McAllister's excellent through ball found Cesnauskis inside the box and the midfielder lofted the ball over keeper David Marshall and into the far corner of the net with a deft chip from an acute angle.
However, a Burchill shot from eight yards that was blocked on the turn was the closest Hearts came to forcing a replay.
Hearts: Gordon, Neilson (Cesnauskis 45), Pressley, Webster, McAllister, MacFarlane (Wyness 77), Hartley, Hamill (Kizys 58), Wallace, Burchill, Miller.
Subs Not Used: Moilanen, Berra.
Goal: Cesnauskis 60.
Celtic: Marshall, Varga, Balde, Valgaeren, McNamara, Lennon, Petrov, Sutton, Thompson, Bellamy, Hartson (McGeady 72).
Subs Not Used: Douglas, Henchoz, Lambert, Maloney. Booked: McGeady.
Goals: Sutton 3, Bellamy 49.
Ref: S Dougal