Clyde's work-rate before the break kept Celtic away from the danger zone although it looked as though the visitors were keeping a gear in reserve.
When Celtic finally broke through, it was thanks to the height advantage in the box. Juninho swung over an accurate corner and Stan Varga guided his header past Bryn Halliwell.
The Bhoys replaced Chris Sutton with Shaun Maloney at the start of the second half and his first taste of the action was when he was upended in the box. Alan Thompson duly converted the spot kick to put the visitors two ahead.
It was nearly all Celtic after that and Halliwell denied Juninho with a full-length save before Stilian Petrov hit the bar with the keeper well beaten.
Another goal had to come and it was Juninho who provided the opening for Petrov to feature on the scoresheet.
This was followed by a Thompson free-kick which was just a fraction too high and the same player deceived the Clyde defence with a corner which found the unmarked Varga at the far post for an easy tap-in.
Celtic completed the scoring with a goal from Craig Bellamy after the Welshman had been set up by neat work between Juninho and Maloney.
The visitors saved Clyde from further embarrassment by indulging in some exhibition stuff in the closing minutes.