Last season's Europa League finalists Ajax went out in the qualifying rounds as Rosenborg beat them to reach the group stage.
The Dutch giants trailed 2-0 on aggregate when Nicklas Bendtner scored.
But they led on away goals after Amin Younes and Lasse Schone both scored in the space of two second-half minutes.
However, Samuel Adegbenro - who also fired home in the first leg - netted twice in the last 10 minutes to send the Norwegians through.
Ajax, who lost to Nice on away goals in their Champions League qualifier earlier this summer, have failed to win a European game in a season for the first time since 1990-91, when they did not compete.
AC Milan beat Shkendija of Macedonia 1-0 to complete a 7-0 aggregate win and qualify for the group stage after three years out of European competition.
Marseille went through 4-1 on aggregate after a 3-0 home win over Slovenians Domzale.
Athletic Bilbao are also in the groups, after beating Panathinaikos 1-0 and 4-2 overall.
Vardar became the first Macedonian side to reach the group stage of a European competition after a 2-1 win at Fenerbahce, secured a 4-1 aggregate victory.
- 21MidtsjøBooked at 21mins
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