The Norwegian side won 3-1 on the night to progress to the next qualifying stage 4-3 on aggregate.
The win was the first game under new manager Kare Ingebrigtsen following the departure of Per Joar Hansen on Monday.
Ingebrigtsen after the game said to newspaper VG: "Simply a fantastic start."
Sligo carryied a first leg advantage after their surprise 2-1 away win earlier in the month. The Irish side went a goal ahead after just 13 minutes of the second leg. Danny North scored after an attacking start to the game from Sligo.
But the Norwegian team returned to form and were the dominant team, countering every Sligo attack and scoring three goals to go through to the next stage of the tournament.
Pal Andre Helland levelled the tie for Rosenborg, before goals from Danish international Mike Jensen in the 49th minute and Finnish striker Riku Riski sealed victory in the 65th minute.
Rosenborg now progress to the third round of the Europa League and face Turkish side Kardemir Karabukspor.
Ingebrigtsen said about tthe next game to NTB:"I don't even know where that is. We'll now go home, train, and then there's the match against Molde. That is the one we're thinking of now."