The clubs agreed a £44.3m fee for the transfer of the Argentine last weekend and United confirmed the deal had been done on their Twitter feed.
United tweeted: “#mufc can confirm the sale of Angel Di Maria to Paris Saint-Germain is complete. We wish him well for the future.”
Di Maria said: “I am very proud, and can’t wait, to wear the colours of Paris Saint-Germain.
“I want to win every competition we enter, both in France and Europe. I want to win the Champions League, which is a dream for all footballers, for a second time.
“We will do everything we can to bring Europe’s biggest trophy to France and the fans of Paris Saint-Germain.”
PSG president Nasser Al Khelaifi said: “We are very proud that Angel Di Maria has chosen Paris Saint-Germain.
“He is an incredibly talented player, with a frightening speed and ability to beat opposition players. He adds his name to the club’s strong South American community and, above all, the big Paris Saint-Germain family.
“By signing Angel Di Maria, Paris Saint-Germain is once again showing the club’s constant ambition to reach the summits of the European game.”
United signed Di Maria from Real Madrid for a British record £59.7m last summer, but after an impressive start, his form dipped and he only started one of the final 10 games of last season.