The Hoops played without a recognised centre forward due to injury and suspension — midfielder Tom Rogic was ostensibly the furthest man in attack — and they missed a focal point in an encounter where the visiting keeper Andre Hansen had little to do.
Indeed, the Norwegian side, who needed extra-time to get past Dundalk in the previous qualifier, created the better chances and Celtic keeper Craig Gordon made several saves to give the Scottish champions, big favourites for the tie, at least a clean sheet to take into the second leg.
Celtic dominated the early possession but it became clear that the lack of a spearhead was proving problematic, despite some nice build-up play.
Midfielder Stuart Armstrong had a shot blocked in the ninth minute after drifting into the box to take a delightful Olivier Ntcham pass and Rogic just could not get the loose ball under control.
The Eliteserien leaders, 17 games into their season, threatened in their occasional forays and in the 21st minute Milan Jevtovic got behind left-back Kieran Tierney, his angled drive blocked by Ajer for a corner which came to nothing before Gordon turned over an attempt by Anders Konradsen from 14 yards.
Vegar Eggen Hedenstad’s whipped-in free-kick early in the second half was tipped over by Gordon. Rosenborg should have taken the lead in the 61st minute when a loose pass by skipper Scott Brown was taken by Nicklas Bendtner but when Jevtovic eventually set up Yann Erik de Lanlay 12 yards from goal, he blasted over.
Seconds later, inside a nervy Celtic Park, Bendtner forced Gordon into a decent save with a drive and at the other end, as the Hoops upped the tempo, James Forrest drilled a shot inches past the far post.However, in another Rosenborg attack, captain Mike Jensen set up Bendtner 12 yards out but he missed his kick completely, ensuring another let-off for the Glasgow side.
Gordon blocked a drive from substitute Elbasan Rashani in injury time before the final whistle confirmed that Celtic still have a bit to do next week if they are to reach the coveted group stages.
Gordon, Lustig, Simunovic, Ajer, Tierney, Ntcham (Hayes 63), Brown, Forrest (Benyu 81), Sinclair, Rogic, Armstrong.
De Vries, Bitton, McGregor, Ralston, Eboue.
Hansen, Hedenstadt, Reginiussen (Bjordal 65), Skjelvik, Meling, Midtsjoe, Konradsen, Jensen, Jevtovic (Rashani 79), Bendtner, de Lanlay (Vilhjalmsson 80).
Ostbo, Rasmussen, Gersbach, Lundemo.
Tiago Martins (Portugal).