Having brushed aside Alashkert 6-0 on aggregate in the previous round, Celtic knew they faced a much tougher test when Birger Meling gave Rosenborg the early lead at Parkhead.
Meling coolly slotted past Craig Gordon from 12 yards and the defender almost doubled his tally as the visitors started strongly.
Slowly Celtic worked their way into the match and after Callum McGregor had tested Andre Hanse they levelled just before half time.
Odsonne Edouard ensured parity at the break with a composed finish after excellent work from Scott Sinclair on the left, the winger cutting in and laying it on a plate for the French striker.
Celtic's second came a minute after the restart when Olivier Ntcham fired home from outside the penalty area, and when Edouard doubled his tally with a delicate chip victory was assured.
There was still time for Jack Hendry to hit the woodwork as Celtic finished with a flourish as Brendon Rodgers' side secured a commanding lead heading to Norway next week.
AEK Athens lie in wait for the victors of this tie, with Celtic now favourites to seal a meeting with the Greek side.