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Goals from Marquinhos and Angel Di Maria put PSG 2-0 up at the interval and Juan Bernat added the third in the 56th minute for a comfortable 3-0 win over Leipzig.
Paris Saint-Germain continued their dominance in domestic competitions as they beat Lyon 6-5 in a penalty shootout.
Bayern Munich won the Champions League for a sixth time by beating Paris St Germain 1-0 in the final thanks to a goal from former PSG youth academy graduate Kingsley Coman.
Paris St Germain scored two late goals, including a stoppage-time winner from substitute Eric Choupo-Moting, to beat outsiders Atalanta 2-1.
Neymar and Juan Bernat scored before half-time to secure a 3-2 aggregate win for PSG after a game played behind closed doors due to the coronavirus.
United had never previously lost at home in European competition by a margin of more than one goal and history suggests their task in the second leg is extremely tough.
Despite the recent slump, PSG secured the Ligue 1 title last month and remain 17 points clear at the top of the table, while Nice are seventh on 52 points.
The runaway champions can clinch a domestic double when they take on Stade Rennais in the French Cup final next Saturday.
The Brazilian forward joined PSG from Barcelona on a five-year deal for a world record transfer fee of 222 million euros ($249 million) in 2017.
The 27-year-old prompted more speculation with a brief Instagram video, showing a black-and-white image of himself in a shirt with the badge of Barcelona and a quote from the Bible.
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