Kieran Tierney gave Brendan Rodgers' side an early lead and Odsonne Edouard grabbed the winner in 79 minutes, just after the visitors had equalised.
With two games still to play, Celtic are level on points with RB Leipzig in Group B while Salzburg have a 100% record and Rosenborg are yet to get off the mark.
Earlier in the evening, Rangers suffered their first defeat in Group G, losing 4-3 to Spartak Moscow in Russia. Steven Gerrard's side remain well in contention to qualify and are just a point behind leaders Villarreal.
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