Any chance of an upset disappeared within the opening ten minutes when Maciej Zurawski scored twice.
At that stage Dumbarton must have feared a heavy defeat, but as it was Celtic had to wait until just before half-time to score their third through Jan Vennegoor of Hesselink.
The Bhoys were only able to add one further goal to their tally in the second half through substitute Derek Riordan.
Dumbarton kept trying to pass the ball throughout the match and can be proud of their efforts. Their only disappointment will be their share of the gate money as a crowd of just under 19,000 turned up.
The visitors' chance of glory came in the opening minute of the game when a cross by Ryan Borris dropped over Lee Naylor but, though Paul McQuilken's shot bounced over keeper Artur Boruc, it rolled narrowly past the post.
Celtic took the lead when Shunsuke Nakamura's long pass found Zurawski behind the Dumbarton defence and the Pole coolly lobbed his shot over keeper Stephen Grindlay.
The home side quickly added a second when Zurawski stepped in front of Andy Geggan to shoot low past the keeper.
Grindlay then produced an excellent double save to deny Vennegoor of Hesselink. The keeper deserved his luck minutes later when Zurawski rounded him after a mistake by David Craig but could only hit the post from a tight angle.
Celtic went three up just before half-time. From Zurawski the ball broke to Aiden McGeady and Grindlay did well to turn his shot on to the bar, only for it to fall to Vennegoor of Hesselink who touched the ball home from close range.
With the game won Celtic took the opportunity at the start of the second half to rest Nakamura and bring on Derek Riordan.
From the substitute's corner Celtic nearly added another goal when Vennegoor of Hesselink slid the ball through to Zurawski whose shot on the turn was parried out by Grindlay.
The home team finally got their fourth goal when Riordan scored from six yards after a clever pass by Zurawski.
There was no further scoring but still time for the Dumbarton keeper to produce a fabulous save to deny Naylor and for McGeady to hit the post with a curling shot from 20 yards.