Ernesto Valverde's team predictably dominated the opening 30 minutes, but PSV looked risky on the counter-attack with Steven Bergwijn curling a shot just wide of Marc-Andre ter Stegen's goal and Hirving Lozano missing one from distance after a speedy run down the right flank.
PSG had come back from two goals down with Kylian Mbappe equalising in the 83rd minute after Thomas Meunier's 40th minute strike halved the deficit.
Regarded by many as the greatest footballer of our generation, arguably, Messi has done things on the pitch that other footballers can only dream of, if not execute on a video game.
But Mauro Icardi's brilliant late volley brought the hosts level, and Matias Vecino headed in a stoppage-time victor for Inter, back in the Champions League for the first time since 2011-12.
Julie Chen Is Officially Leaving CBS' The Talk After Les Moonves Controversy
Sara Gilbert also weighed in, saying that while she loves and supports Chen "always", this is "an important time in our culture". Chen will address her exit, which is effective immediately, in a videotaped message that will air on Tuesday's episode.
PSG next face Red Star Belgrade, who marked their return to the big time with a 0-0 draw against Napoli.
Barca have taken a backseat to arch rivals Real Madrid in the Champions League in the last three seasons, last term's stunning loss to Roma marking their third successive quarter-final exit, and for long periods they offered little to suggest a challenge is forthcoming this season.
Goalkeeper Milan Borjan made a flurry of telling saves while Napoli's Lorenzo Insigne rattled the crossbar in the first half and Jose Callejon had an effort cleared off the line after the break.
Samuel Grandsir scored from close range to put the hosts ahead in the 18th minute. However, Atletico dominated for long spells and struck back before the interval through goals from Diego Costa and Jose Gimenez. "This was typical of the way Messi has been playing the last two, three years".
"I would like to be much more effective on penalty kicks".