All hope appeared lost for Celtic, who were outplayed for large stretches, after Munas Dabbur and Fredrik Gulbrandsen struck twice in 10 second-half minutes.
But the home fans were in boisterous sprits in the closing moments as news filtered through of the result in Germany and they had additional reason to cheer when Olivier Ntcham reduced the deficit in stoppage time.
The hosts rode their luck throughout the opening 45 minutes with Diadie Samassekou and Stefan Lainer both hitting the woodwork.
Dabbur, twice on target in the reverse encounter between these teams, finally made Salzburg's dominance count by heading in Lainer's cross in the 67th minute.
Craig Gordon gifted away a second 10 minutes later as Gulbrandsen stole the ball from the goalkeeper's throw and tucked into an empty net, but the blow was softened by Rosenborg's unexpected late goal against Leipzig.
Jerome Onguene then fouled Filip Benkovic deep into stoppage time and Ntcham was able to add a footnote to a strange match from the spot, the midfielder pouncing on the rebound of Alexander Walke's save from his initial attempt.