Fredrik Gulbrandsen scored the only goal of the game for the Austrian side at the Red Bull Arena Salzburg.
That means Salzburg have won Group B and will have nothing to play for when they travel to Parkhead in a fortnight’s time.
Leipzig trail Celtic by three points with one game remaining. They will need to defeat Rosenborg in Germany and then hope Salzburg do them a favour in Glasgow.
Celtic were able to push themselves into second place in the group as Scott Sinclair’s first-half header was enough to defeat Rosenborg.