Brendan Rodgers' men left it late to get the better of Rosenborg in the opening round of matches before being beaten 3-1 in Salzburg last time out.
They were unable to get back to winning ways in Leipzig, however, as Matheus Cunha and Bruma scored in the opening half for the hosts.
Cunha had been threatening to find the net before he eventually broke the deadlock after 31 minutes, Celtic failing to deal with a Marcel Sabitzer cross before giving the Brazilian forward the time and space to fire home from the centre of the penalty area.
Bruma's goal followed four minutes later as he reacted quickly to punish some more suspect defending from the Scottish champions.
Combined with their costly defensive lapses, Celtic offered little going forward, and they almost fell three behind when Jean-Kevin Augustin had a fierce strike tipped onto the post by Craig Gordon after 65 minutes.
The result lifts second-placed Leipzig three points clear of Celtic in Group B after three matches, while Salzburg are a further three points ahead of the German club following their 3-0 victory over Rosenborg on Thursday.