After a 1-1 draw in a tense first leg in the Netherlands, Kevin-Prince Boateng scored twice and Mario Balotelli added another to ensure Milan won the playoff 4-1 on aggregate and advanced to Thursday's draw.
At Glasgow, Celtic beat Shakhter Karagandy 3-0 to overcome a 2-0 loss in the first leg. James Forrest's stoppage-time goal gave the Scottish champions a 3-2 win on aggregate. Real Sociedad and Zenit St. Petersburg also advanced Wednesday along with Plzen, which progressed after beating Maribor 4-1 on aggregate.
Milan, which had a surprise 2-1 defeat to promoted Hellas Verona last weekend in its Serie A opener, qualified for the group stage for the 11th time in 12 seasons, and 17th overall.
Boateng unleashed a shot from outside the box into the left-hand corner to put the home team ahead in the ninth minute. Stephan El Shaarawy then came close but saw his shot hit the crossbar
Balotelli scored his 13th goal for AC Milan since his move from Manchester City with a volley from close range in the 55th after Phlippe Mexes flicked on a corner from the right.
"We're happy. We played a great match. We also played well in the first leg," said Balotelli, who could not play for Milan in the Champions League last season after a January transfer. "It's the Champions League and it means a lot to us to do well."
Boateng scored his second goal of the evening after collecting the ball on the left side of the box and firing a shot into the far corner in the 77th.
Celtic will play in the group stage for the eighth time after Neil Lennon's team produced one of its trademark comebacks. Celtic erased the first-leg deficit with goals from Kris Commons and Giorgos Samaras on either side of halftime before Forrest sealed the home team win from Anthony Stokes' cross.
"It's the best feeling ever, we are in the Champions League, where we should be," Forrest said. "We've done it in 90 minutes and that is what we wanted."
Real Sociedad will play in the group stage for the first time in 10 years after Carlos Vela scored twice in a 2-0 win over Lyon. The French side had the upper hand in the first half but failed to create real chances and bowed out from the tournament 4-0 on aggregate.
Zenit booked its spot in the group stage after beating Pacos Ferreira 4-2 for an 8-3 aggregate. Portugal playmaker Danny scored twice for Pacos before Aleksandr Bukharov and Andrey Arshavin added goals to ensure their Russian club qualified for the Champions League for the fourth time.
Arsenal also qualified via the playoffs with a 5-0 win on aggregate over Fenerbahce on Tuesday, alongside Schalke, Steaua Bucharest and Basel.