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The violence occurred around the Parc des Princes stadium and on the Champs-Elysees avenue during and after the match on Sunday night, police said on Twitter.
Thousands of supporters had gathered at the Parc des Princes to cheer on PSG, lighting flares and chanting as they watched the 1-0 defeat in Lisbon on a big screen.
The trouble lasted several hours, as groups of fans threw bottles and fireworks at police, who responded with tear gas and rubber bullets, an AFP reporter said.
On the Champs-Elysees, vehicles were set on fire, windows broken and shops vandalised.
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Police said 148 people were arrested "in particular for damage, violence or throwing projectiles".
Interior Minister Gerald Darmanin said 16 police were injured, 12 shops attacked and about 15 vehicles damaged overnight.
In a tweet on Sunday night, he condemned the "unacceptable" violence and thanked the police who arrested "small violent groups, troublemakers."
Social distancing measures were ignored inside and outside the Parc des Princes among fans who had hoped to see their Qatar-owned team's first triumph in Europe's top club competition.
Police said 404 people were stopped for not wearing face masks as part of regulations aimed at curbing the coronavirus outbreak.
Paris mayor Anne Hidalgo had earlier called for "mature celebrations" should PSG beat Bayern, after supporters clashed with police following their 3-0 win over RB Leipzig in the semi-finals.
"I'm gutted! They wasted too many chances and unfortunately in the final that just doesn't fly," said Nicolas Mounier as he left the stadium peacefully.
"We are disappointed but we weren't dreadful," another PSG fan, Anne Vaneson, told AFP.
"In the first half, we were on a level playing field but we paid for a dip in performance in the first 20 minutes of the second half."
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