PSG set up Club World Cup semi clash with Mbappe's Real Madrid
Los Blancos squeezed out Borussia Dortmund 3-2 in New Jersey in the day's other quarter-final clash, with Mbappe on the scoresheet for the first time at the tournament.
Luis Enrique's PSG needed to dig deep in front of 67,000 fans at the Mercedes-Benz Stadium after late red cards for Willian Pacho and Lucas Hernandez.
The French side won the UEFA Champions League for the first time in their history in May, despite their key player Mbappe departing for Real last year.
Young forward Desire Doue, who shone on PSG's road to European glory, and Ballon d'Or candidate Ousmane Dembele got the goals in the second half of an enthralling clash against Vincent Kompany's Bayern, who had two strikes ruled out for offside.
The German giants also lost playmaker Jamal Musiala to a horror ankle injury just before half-time, which Kompany said made his "blood boil" because of how unfortunate it is for the 22-year-old.
"After the first red card it became complicated, then with two players less it was a bit of a lottery," said a relieved Luis Enrique.
PSG goalkeeper Gianluigi Donnarumma and his Bayern counterpart Manuel Neuer both made several excellent saves in a pulsating battle at the air-conditioned, covered venue which allowed both sides' high-pressing, attacking approaches to shine.
Players and spectators at both matches observed a minute of silence in honour of Liverpool forward Diogo Jota and his brother Andre Silva, who tragically died in a car crash on Thursday.
PSG's clash with Bayern was FIFA's expanded tournament's biggest heavyweight clash thus far and the game delivered, even if fans had to wait until the 78th minute for the opener.
After Harry Kane gave the ball away, PSG surged forward and 20-year-old Doue fashioned some space on the edge of the box before flashing a low strike home at the near post.
Bayern were given hope as they hunted for an equaliser when PSG defender Pacho was sent off for a high tackle on Leon Goretzka after 83 minutes.
Kane had a goal disallowed for offside, like Bayern's Dayot Upamecano in the first half, and after Hernandez was dismissed for an elbow Dembele tucked home the second in stoppage time, paying tribute to Jota by copying his video-gaming celebration.
"In the first half we showed them who we are and they had problems with us," said Bayern substitute Thomas Mueller, who played his last game for the club.
Mbappe strikes
Real Madrid cruised for the most part against Dortmund before a flurry of goals at the end left them in danger of losing their lead.
Xabi Alonso, unbeaten at the helm, shifted back to a four-man defence against Niko Kovac's side, shorn of suspended Jobe Bellingham, who would have loved to face his older brother, Madrid's Jude.
The coach decided to leave Mbappe, who missed the three group games after a stomach bug, on the bench and persist with Gonzalo Garcia up front.
The 21-year-old striker paid back Alonso for his confidence by grabbing his fourth goal of the tournament after 10 minutes with a close-range finish from Arda Guler's cross.
"He's doing great work for the team... he's in the right position in the box, he's a proper number nine," said Alonso.
In the 20th minute Fran Garcia doubled Los Blancos' lead, rushing into the box to polish off Trent Alexander-Arnold's dangerous ball across the face of goal.
Madrid were cruising to victory until Max Beier drilled home in stoppage time.
Real responded, with substitute Mbappe netting his 44th goal of the season with an acrobatic finish from another Guler cross.
He paid tribute to the late Jota by making the number 20 with his hands, the Portuguese striker's shirt number at Liverpool.
Madrid were not home and dry. Dean Huijsen was sent off for tugging back Serhou Guirassy as he ran through and the Guinea international netted from the spot, but Dortmund had no time to equalise.
"Everything was under control with the result, not conceding much, but the last 10 minutes were kind of crazy," said Alonso.
"We lost our shape and our intensity, and luckily nothing went wrong."