The Norwegian club took a 1-0 advantage into the home leg of their play-off clash thanks to recent signing Adegbenro also being on target in Amsterdam.
Former Arsenal striker Bendtner doubled the advantage for Kare Ingebrigtsen's men when he headed home at the far post in the 26th minute.
Ajax appeared to be heading through on the away goals rule when Amin Younes and Lasse Schone struck within a minute of one another as the game ticked past the hour but Adegbenro had other ideas.
The 21-year-old Nigerian pulled Rosenborg level on the night with 10 minutes to play and put the tie to bed as he surged through a beleaguered Ajax defence a minute from time to net a stunning solo effort.
Australia left-back Gersbach watched the action from the bench.
Adegbenro was outdone when it came to the goal of the night, however, as Gylfi Sigurdsson marked his full Everton debut by volleying home improbably from 50 yards at Hajduk Split.
That ensured a 1-1 scoreline on the night, after Josip Radosevic's excellent first-half opener, and a 3-1 aggregate triumph for the Premier League side.
AC Milan youngster Patrick Cutrone continued his stellar start to the season by scoring after 13 minutes against Shkendija, although there would be no repeat of last week's 6-0 hammering as the home side closed out a 7-0 aggregate defeat.
Shkendija's compatriots Vardar claimed a stunning upset as they won 2-1 in Turkey to dispatch Fenerbahce 4-1.
Roman Neustadter gave Fener hope in the 61st minute but Jaba Jigauri reasserted Vardar's control seven minutes later and substitute Nikola Gligorov completed a memorable triumph with a breakaway goal in stoppage-time.
Former Monaco forward Valerie Germain notched a brace as Marseille made light of a their first-leg struggles and a 1-1 draw against Domzale with a 3-0 victory that was crowned by Florian Thauvin's overhead kick.
Athletic Bilbao made sure of progress against Panathinaikos as Iker Muniain slid in to crown a wonderful team move midway through the first half – the only goal at San Mames as Jose Angel Ziganda's men prevailed 4-2 overall.