The home side had taken a 3-0 lead through Angelino, Amadou Haidara and substitute Justin Kluivert, with the game looking all but over.
United never give in though, and a Bruno Fernandes penalty and Paul Pogba header from a corner gave us hope late on.
The Reds almost found a dramatic equaliser in the dying seconds, but Leipzig held firm, meaning we're now in the last 32 of the Europa League instead.