Holding a 3-1 lead after last week's quarterfinal first leg, PSG built a four-goal cushion thanks to goals by full backs Achraf Hakimi and Nuno Mendes inside 27 minutes at Villa Park on Tuesday.
But Villa, cheered on by Prince William, roared back thanks to a deflected shot before halftime by Youri Tielemans and then goals in a two-minute span from John McGinn and Ezri Konsa by the 57th.
Only a string of fine saves from PSG goalkeeper Gianluigi Donnarumma denied Villa a fourth goal that would have taken the match to extra time.
PSG advanced 5-4 on aggregate to knock out a second Premier League team, after Liverpool in the round of 16.
Next up could be another in Arsenal, which leads Real Madrid 3-0 from the first leg.
The French champions advanced to the semifinals for the third time in five seasons and this time without their "galacticos," with Neymar, Lionel Messi and most recently Kylian Mbappé leaving the club.
In Tuesday's other quarterfinal, Serhou Guirassy scored a hat trick and still ended up on the losing side as Borussia Dortmund's 3-1 win wasn't enough to prevent a 5-3 aggregate loss to Barcelona.
After being swept aside 4-0 in the first leg, Dortmund had nothing to lose and attacked Barcelona from the start, leaving the visitors off-balance.
Guirassy had missed chances in the first leg and spurned a couple more early in Tuesday's game but made no mistake from the penalty spot in the 11th minute after Barcelona goalkeeper Wojciech Szczesny collided with Pascal Gross.
Guirassy's header made it 2-0 in the 49th minute before an own goal by Ramy Bensebaini made Dortmund's task even harder. Bensebaini's own goal was a double disappointment because he had been crucial in the comeback attempt, assisting Guirassy's second goal shortly before.
Guirassy completed his hat trick with a powerful close-range shot in the 76th, in the process becoming the Champions League's top scorer this season on 13 goals.
It gave Dortmund fresh hope of completing what would have been one of the all-time great Champions League comebacks, but Barcelona held on to progress.
Barcelona faces either Inter Milan or Bayern Munich in the semifinals.
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