Mbappe's smart close-range finish cancelled out Lionel Messi's penalty opener, before he tapped in after half-time to give PSG the lead for the first time.
Moise Kean added to that advantage before a sublime breakaway fourth saw Mbappe sweep a stunning finish into the top-right corner.
The France forward is the first visiting player to score three times at Camp Nou since Diego Forlan's hat-trick for Villarreal in 2005.
That Forlan effort was the first of the 21st century, the previous being the only other Champions League treble besides Mbappe's, when Andriy Shevchenko led a remarkable 4-0 away win for Dynamo Kiev in 1997.
Not all hat-tricks for away sides at Barca have resulted in such dominant displays, however.
Forlan could only secure a 3-3 draw, with three of the 10 trebles coming in matches that ended all square.
And two three-goal stars actually finished on the losing side, including poor Milinko Pantic in a Copa del Rey epic months before Shevchenko stole the show.
Pantic scored four times for Atletico Madrid, yet they improbably went down 5-4 to Bobby Robson's Blaugrana.
The only other instance of a visiting player netting four came at Barca's previous Les Corts home when Julian Arcas did so in a draw with rivals Espanyol in 1956.
Two of the 10 hat-tricks came at Les Corts - the other eight were all at Camp Nou - as Barca also beat Valencia 4-3 in 1948 despite Vicente Segui's treble.
Full list of away hat-tricks against Barcelona:
Vicente Segui (Valencia) - 4-3 defeat at Les Corts in 1948
Julian Arcas (Espanyol) - 4-4 draw at Les Corts in 1956
Tony Kurbos (Metz) - 4-1 win at Camp Nou in 1984
Manuel Pena (Valladolid) - 4-2 win at Camp Nou in 1987
Baltazar de Morais (Atletico Madrid) - 3-3 draw at Camp Nou in 1989
Francisco Higuera (Zaragoza) - 5-4 win at Camp Nou in 1994
Milinko Pantic (Atletico Madrid) - 5-4 defeat at Camp Nou in 1997
Andriy Shevchenko (Dynamo Kiev) - 4-0 win at Camp Nou in 1997
Diego Forlan (Villarreal) - 3-3 draw at Camp Nou in 2005
Kylian Mbappe (Paris Saint-Germain) - 4-1 win at Camp Nou in 2021