The hosts, who needed to win to qualify for the knock-out stages, took a very early lead on two minutes through danger-man Angelino, who fired home at the back post.
Amadou Haidara added a second, on the volley, 11 minutes later and Leipzig had further chances to increase the advantage, including a disallowed goal by VAR for a clear offside.
After Bruno Fernandes hit the bar with a brilliant free-kick on 68 minutes, the home team claimed a third when a deflected cross was allowed to travel all the way through to substitute Justin Kluivert to compound United's frustration.