Some 22,000 police were deployed across France for the game, including 8,000 in Paris, after street unrest marred PSG's win in the competition last year.
Before the match had even ended, police said 2,216 people had been stopped and 89 fined. Forty-five people were arrested, of whom 13 were taken into custody.
Two dozen flares and about 100 fireworks were seized, while a bus shelter was destroyed near the Champs-Elysees avenue in central Paris. One police officer was injured.
Fireworks rang out across Paris after the French side completed their penalty shootout win.
Police said a bakery and a restaurant were damaged near PSG's Parc des Princes stadium, where between 4,000 and 5,000 people gathered during the match and projectiles were thrown at officers.
Near the stadium, about 150 people "attempted to enter through one of the gates" but police pushed them back, a police spokesperson said.
An AFP reporter at the scene said clashes broke out between police and supporters near the stadium, and officers responded with tear gas when fireworks were thrown at them.
Interior Minister Laurent Nunez said there was a "very robust, very solid system in place" to curb violence.
"Our responsibility is to guarantee everyone a festive celebration that is calm and fully secure," a police spokesperson said.
Last year, supporters ransacked shops on the Champs Elysee and other streets after the PSG win. 5,400 officers were deployed around Paris and 491 of the 563 arrests across France were in Paris. Some 307 people were taken into custody, 202 of them in Paris.