Leigh Griffiths proved to be the difference as Celtic sealed their place in the League Cup semi-final at the expense of Hearts.
The former Hibernian striker opened the scoring as he skipped past two defenders before planting his shot into the bottom corner of Neil Alexander’s goal.
He then surged to the byeline and set up substitute Tom Rogic for a second with a simple tap-in to kill off the Jambos.
Arnaud Djoum grabbed his first goal for the club, heading in late for the hosts but it proved to be no more than a consolation.
The result eases the pressure on under-fire boss Ronny Deila and keeps the Hoops’ treble dreams alive.
Their preparation for this game was thrown into disarray minutes before the start when captain Scott Brown pulled up in the warm up and James Forrest took his place.
It meant a tactical reshuffle for Deila but the Hoops started stronger and the first chance fell to Forrest but his goal-bound shot was blocked by his own team-mate Griffiths.
Hearts found possession hard to come by in the opening stages but Osman showed a statement of intent as held off the defender before dragging his shot wide, McGhee also had a couple of efforts.
Celtic missed a glorious opportunity just before half-time when Stuart Armstrong had a fresh-air swipe from around eight yards.
After the break Hearts came out with a bit more attacking impetus and Osman Sow thought he’d put the hosts ahead turning in Paterson’s knock-down from close range but celebrations were cut short by the linesman’s flag.
However, there was no doubt about Celtic’s opener as Griffiths, who had played on the periphery, suddenly took centre stage, skinning a couple of Hearts defenders before firing the ball low past Alexander.
It was a moment of brilliance which had been lacking all game and Griffiths was instrumental in the second, too, driving to the byeline and squaring the ball back for Rogic who tapped in from close range.
Hearts halved the deficit near the end when Djoum planted a header past Craig Gordon but it proved to be too little, too late.