Vincent Marcel grabbed a 79th-minute equaliser for Nice after Ajax's Donny van de Beek and Davinson Sanchez had cancelled out Arnaud Souquet's early opener for the French side.
Celtic had only managed a scoreless draw at home to Rosenborg and struggled to break down the Norwegians in Wednesday's second leg.
James Forrest, however, finally found the net in the 69th minute, firing home from a tight angle to put Brendan Rodgers's side through with a 1-0 victory.
There was plenty of drama in Berne as Swiss side Young Boys recovered from a 3-1 defeat away to Dynamo Kiev, winning 2-0 at home to go through on away goals.
FC Copenhagen hammered FK Vardar 4-1 at the Parken stadium in the Danish capital to go through 4-2 on aggregate, while Turkish league runners-up Istanbul Basaksehir cruised past Belgian side Club Bruges 2-0 at home to win 5-3 on aggregate.
The successful teams now go forward to the playoff round, with the winners qualifying for the group stage of Europe's lucrative elite club competition.
The draw for the playoff round takes place on Aug. 4.
(Reporting by Philip O'Connor in Oslo; Editing by Toby Davis)
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