Henrik Larsson headed his 43rd goal of the season to send Celtic into the last four of the Scottish Cup and bring their historic Treble dream one step closer.
The super Swede now has Jimmy McGrory's all-time scoring record of 50 goals in season 1936-37 and Charlie Nicholas' post-war milestone of 48 in season 1982-83 firmly in his sights.
But more importantly for Larsson and Martin O'Neill was the victory which saw the 1995 winners take on Dundee United in the semi-final while Hibernian play Livingston. "I thought his goal fantastic, an absolutely brilliant goal.
"No matter how great the ball is in you still have to put it in the back of the net so that was fantastic - he is a truly top-class player," said O'Neill. With a 13-point lead at the top of the Premier League and next week's CIS Insurance Cup final against Kilmarnock to look forward to, Hearts were hoping to cause an upset.
But the Edinburgh outfit, who had conceded 18 goals to Celtic before yesterday, never deserved anything other than a defeat.
Hearts offered little in the first-half and the only surprise was that it took the home side 41 minutes to break the deadlock.
As expected Celtic started confidently, knocking the ball around and they had Hearts on the back foot in the opening exchanges.
Alan Thompson was finding plenty of space on the left flank and in the fifth minute Robert Tomaschek had to head behind with Didier Agathe poised to score.
Celtic did not seem to mind the fact that their fans had stayed away in the droves - with the game shown live on television - and they continued their bombardment.
Boyack and Mjallby both went in the book for reckless tackles in the space of a minute, but Celtic's pressure told in the 41st minute.
Thompson again found space on the right and whipped in an inviting cross for Larsson to head on target and despite Niemi's touch the ball bedded itself in the top corner.
Hearts had to and did come out fighting after the break and Juanjo in particular was causing the Celtic defence major problems on the break.
Hearts were given hope of salvaging something from the game when Johan Mjallby left the field to have stitches inserted into a head wound.
Celtic defended frantically at times and O'Neill eventually was forced to bring on Tom Boyd in place of the brave Swede.
Goalkeeper Robert Douglas was barely tested all afternoon, but he was lucky not to get punished in the dying seconds.
He came off his line, but turned his back on the on-rushing Gary McSwegan, but somehow got a vital touch to deny the visitors.
Celtic: Douglas, Valgaeren, Vega, Mjallby (Boyd 76), Agathe, Lambert, Lennon, Petrov, Thompson, Moravcik (Johnson 66), Larsson. Subs Not Used: Gould, McNamara, Crainey. Booked: Mjallby. Goals: Larsson 40. Hearts: Niemi, Flogel, Pressley, Severin, Tomaschek, McCann, Cameron, Makel (McSwegan 81), Boyack (Murray 46), Juanjo, Kirk. Subs Not Used: McKenzie, McAnespie, Neilson. Booked: Flogel, Boyack, Makel, Tomaschek, Severin.
Referee: K Clark (Scotland).
