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 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.
The suspended Leigh Griffiths will be available for the return game at the Lerkendal Stadion in Trondheim next Wednesday if he recovers from a calf injury - Moussa Dembele’s hamstring injury could keep him out until September - and boss Brendan Rodgers will need more firepower from somewhere if his side are to clinch a play-off spot.
With centre-back Erik Sviatchenko failing a fitness test, and injured Dedryck Boyata out long-term, 19-year-old Norwegian Kristoffer Ajer was handed his first start after returning from a loan spell at Kilmarnock last season.
His previous appearance came 12 months ago as a second-half substitute in the 3-0 home win over Lincoln Red Imps
Former Arsenal striker Nicklas Bendtner led the line for Rosenborg, who needed extra-time to get past Republic of Ireland’s Dundalk in the previous qualifier.
With the ultras-style fan group the Green Brigade banned from Celtic Park for two matches after ‘’serious incidents of unsafe behaviour’’ at Celtic’s previous two home games, the rest of the home support took it upon themselves to gee up the Scottish champions but it was tough going.
As expected 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.
Rosenborg manager Kare Ingebrigtsen would have been happy with the goalless first-half and would have been a whole lot happier had Vegar Eggen Hedenstad’s whipped-in free-kick early after the restart gone under the bar and not tipped over by the Scotland keeper.
The Norwegian side should have taken the lead in the 61st minute when a loose pass by skipper Scott Brown was taken by 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 take the next step towards the coveted group stages.