Marquinhos had opened the scoring for the hosts in the Ligue 1 match before scoring an own goal in the second half.
Things went from bad to worse for PSG as Neymar was sent off after two yellow cards in as many minutes.
But Mbappe saved the three points as he netted his 94th-minute penalty.
The result sees PSG remain top of the table and extend their advantage over second-place Lens to eight points while Strasbourg are three adrift of safety.
The hosts were dominant with the ball, having 71% of the possession, but did little with it and looked neutered in the absence of Lionel Messi, who has not yet returned after winning the World Cup.
Second-bottom Strasbourg had caused issues in the first half as Kevin Gamerio forced Gianluigi Donnarumma into a last-ditch save and seemed to be on course for a surprise draw after Marquinhos’ own-goal.
But while Neymar’s sending off sent PSG into an initial period of frustration, it ignited their attack as they battled back to try to turn the game around.
Mbappe was clipped by Gerzino Nyamsi in the box as he raced towards the Strasbourg goal and got the better of Matz Sels from the spot, tucking it into the bottom-right corner.