Boosted by Danny North’s 13th-minute goal, which put them 3-1 up on aggregate, Rovers were still in contention even when the Norwegian giants levelled through Pal Andre Helland three minutes later.
But a brace of excellent second-half goals from Mike Jensen put paid to Sligo’s hopes and redeemed the visitors after their unexpected home loss a week before.
In front of 3,792 fans, Sligo started briskly with blocked shots from John Russell and David Cawley, while centre-back Evan McMillan’s acrobatic overhead kick just cleared the crossbar in the fourth minute.
Sligo took the lead from a polished move that involved six players, with North netting from close range after Alan Keane’s cross was put into the path of the striker by David Cawley.
Although rattled, Rosenborg equalised three minutes later when Helland punished a defensive lapse to fire an unstoppable shot past Gary Rogers.
The Norwegians were then dealt a blow when Helland was forced off through injury after 27 minutes.
Sligo regained their composure but required Seamus Conneely’s goal-line clearance to keep Rosenborg at bay 10 minutes before the break. Substitute Mikkel Diskerud created the chance for Riku Riski and his half-blocked effort appeared to be limping the line over until Conneely’s intervention.
Cawley unleashed a shot on target two minutes short of half-time as Sligo looked comfortable in defence — Evan McMillan and Kalen Spillane were composed — and capable of further hurting Rosenborg through Kieran Djilali.
Rosenborg took the lead after 48 minutes — levelling matters on aggregate — when Jensen finished well following good work from Diskerud, although Sligo’s defending was not clinical enough.
It could have been even worse for Sligo when, just minutes later, Alexander Soderlund went close with a deft header.
Sligo missed a remarkable chance to retake the lead overall in the 54th minute when North hit the side-netting with a clever shot from a John Russell through ball.
Rosenborg went 3-1 up in the 64th minute — meaning Sligo had to score twice — when Jensen netted his second with a smart finish inside the area after two shots were blocked.
Sligo boss John Coleman threw on two young strikers, Sean Maguire and Ruairi Keating, with 20 minutes left in an attempt to force an increasingly unlikely comeback as Rosenborg, with Diskerud influential, remained in control.
G Rogers; A Keane, E McMillan, K Spillane, R Gaynor (S Maguire 70); P O’Conor (R Keating 70), S Conneely, J Russell, D Cawley, K Djilali (D Ledwith 86); D North
A Hansen; M Dorsin, T Reginiussen, S Strandberg, T Mickelsen; PA Helland (M Diskerud 27), M Jensen, O Selnaes; R Riski, A Soderlund (A Sorloth 78), MG Pedersen
Erik Lambrechts (Belgium).