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Jonathan David converted from the spot three minutes into first-half stoppage time to hand Real, who had not lost in their previous 36 matches in all competitions, their first defeat since January.
Every team will play eight games against different opponents in the new format of Europe's elite club competition.
The 34-year-old Spain international has won six Champions League titles with Madrid since joining their youth academy in 2001, as well as four LaLiga crowns, two Copa del Reys.
Madrid secured a 2-0 win against Borussia Dortmund.
A 4-3 aggregate victory sealed Real's progress to a remarkable sixth Champions League final in 10 years and they will bid for a record-extending 15th title against Borussia Dortmund at Wembley on June 1.
Hummels headed home from a corner five minutes into the second half to send the Germans, who won the title in 1997, into their first final in Europe's top club competition since 2013.
PSG, fresh from being crowned Ligue 1 champions and desperate to win their first ever Champions League trophy, found it hard going against a disciplined German defence, especially in the first half with forward Kylian Mbappe largely neutralised.
Bayern, with only the Champions League title left to fight for, had scored twice in four minutes early in the second half through a Leroy Sane strike and Harry Kane penalty to turn the game around before Vinicius' spot kick.
PSG will face Borussia Dortmund in the semifinals after the Germans beat Atletico Madrid.
PSG came into the match unbeaten in their last 27 games in all competitions but were undone by a fired-up Barcelona side looking to return the club to the summit of European football.
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