Scorers: Celtic - Guidetti (24), Commons (57pen), Eckersley (62og)
John Guidetti’s first goal for Celtic had given Ronny Deila’s side a half-time lead but Hearts spurned a glorious chance to alter the balance of proceedings when Osman Sow smashed a 55th minute penalty over the crossbar.
Two minutes later, Celtic doubled their lead when Kris Commons showed how it should be done from the spot before Adam Eckersley’s own goal put the outcome beyond any doubt.
Deila’s men might easily have won by a bigger margin in the end but this was a welcome and much-needed win for the Norwegian.
Hearts will reflect ruefully on key moments which saw their hopes of an upset slip away from them but it was a defeat which should have no impact on the confidence of Robbie Neilson’s men as they retrain their focus on the primary aim of promotion from the Championship.
This tournament has been very much at the bottom of Celtic’s list of priorities in recent seasons, although that only partly explains why they have won it just twice in the last 13 years.
But in the wake of his underwhelming start to life as Celtic manager, Deila could not afford to treat it with anything but the utmost seriousness. He duly fielded as strong a team as possible with Scott Brown and Commons among those restored to the starting line-up.
Guidetti was preferred to Stefan Scepovic as the central striker, supported by an attacking midfield trio of Callum McGregor, Commons and Anthony Stokes, while Brown was joined by fit-again Stefan Johansen in the holding positions.
With the top tier of Celtic Park closed for the night, there was a relatively muted atmosphere inside the stadium despite the best efforts of both the Green Brigade section of the home support and the 1200-strong travelling Hearts fans.
The visitors duly carried the air of a team high on confidence and there were some early signs that Sow was capable of causing the Celtic defence some discomfort. The big Swedish striker, operating in front of an advanced midfield trio of Dale Carrick, Sam Nicholson and Jamie Walker, held the ball up well and dragged the home back four out of position on several occasions.
Jason Denayer was booked for a foul on Sow as early as the third minute as Celtic took some time to find any telling rhythm or tempo of their own. Walker saw a shot deflected wide off Johansen, then Denayer had to scramble the ball behind as he cut out a Callum Paterson cross with Sow lurking menacingly.
But Celtic gradually gathered some momentum and should have gone in front in the 15th minute when a mistake from Alim Ozturk left Guidetti one-on-one with Hearts ‘keeper Neil Alexander. The striker tried to curl his shot into the corner of the net but sent it wide.
It was the spark for a sustained spell of territorial advantage for Celtic which saw Alexander suddenly becoming a busy man. He held a shot from Commons at his left hand post then reacted smartly to beat away a stinging volley from Brown after a slick move from the champions.
There was a growing sense the breakthrough was coming and it duly arrived in the 24th minute. Emilio Izaguirre, causing Hearts all sorts of grief with his raiding runs from left-back, whipped over a cross which Danny Wilson could only head weakly into the path of Guidetti who cracked home a left foot shot from around eight yards.
Hearts left-back Eckersley had to effect a desperate goal-line clearance as Celtic tried to turn the screw, then only Alexander’s fine reaction save prevented Morgaro Gomis from diverting another Izaguirre cross into his own net.
But as the half-time whistle approached, Hearts so nearly equalised. Izaguirre’s woefully short pass-back left Walker racing clear but although he managed to loft his shot over Craig Gordon, the ball struck the crossbar.
The momentum might so easily have shifted in Hearts’ favour had Sow not fluffed his lines so horribly ten minutes into the second half. He went down under Virgil van Dijk’s challenge, referee Willie Collum pointing to the spot without hesitation. Sow stepped up confidently to take the penalty but blazed it wildly over the crossbar.
Just two minutes later, Celtic had a spot-kick of their own and the tie was effectively over. Ozturk fouled Guidetti, again leaving Mr Collum in no doubt, and Commons coolly sent Alexander the wrong way from 12 yards.
Hearts’ evening was now threatening to implode very badly as they went 3-0 down in the 62nd minute. It was a calamity for Eckersley whose ill-advised attempt to chest a Stokes cross back to Alexander was him send the ball into his own net.