The game looked to be drifting to a goalless draw before the Frenchman drove at the Madrid defense and slotted a fine finish past Thibaut Courtois in the 94th minute of the match.
It came after Lionel Messi saw his second-half penalty saved by the Madrid keeper.
PSG would have been frustrated had it not capitalized on its dominance at the Parc des Princes stadium — the host had 21 efforts on goal compared to Madrid’s three and dominated possession.
But despite Messi, Mbappe and eventually Neymar on the pitch, it took a last-gasp effort from the brilliant Frenchman to earn the win.
Mbappe, who has been linked with a move to Real Madrid next season, had been the most threatening player throughout the match with his pace and trickery giving Madrid’s defense nightmares at times.
The 23-year-old now has five goals in this season’s Champions League and, perhaps more importantly, dug Messi out of a rather embarrassing hole.
The Argentine drifted through much of the game but had a golden opportunity to give his side a well-earned win.
Courtois was equal to Messi’s efforts, though, diving to his left to keep the tame penalty from hitting the back of the net.
This season’s knockout stage is different from previous years with the away goal rule being scrapped.
It gave the visitors an incentive to leave Paris with a goalless draw that they could build on at home and it almost got its wish, until Mbappe’s heroics.
Despite the first-half remaining goalless, it was the hosts who dominated.
Mauricio Pochettino’s side didn’t allow Real Madrid to have a moment’s peace on the ball and it managed to isolate its most potent threat of Karim Benzema.
While the Parisians had most of the ball, it couldn’t find a way through Madrid’s defense.
With Messi drifting on the periphery, Mbappe was the star man as he gave Madrid’s right-back Dani Carvajal a torrid 45 minutes.
The Frenchman used his pace and trickery to leave the defender in his wake on more than one occasion but failed to pick out the right pass to create the breakthrough for his side.
The second half started similarly to the first with PSG making all of the play — Mbappe once again the talisman.
He saw a fierce shot expertly kept out by goalkeeper Courtois as the hosts were roared on by the relentless home crowd.
With only 30 minutes remaining, Mbappe once again took aim at Carvajal and this time got his reward after the defender brought him down in the box and the referee pointed to the penalty spot.
Messi was the man to take the ball but saw his spot-kick saved by Courtois.
In search of a goal, Pochettino turned to Neymar who was making his return from injury.
The Brazilian, who had not played since November, got to work quickly, jinking his way past defenders with his first touches of the ball.
With Neymar, Messi and Mbappe on the pitch, Madrid’s defense was forced to dig in but it crumbled at the final moment.
Mbappe, once again, left Madrid’s defenders on the ground and guided his shot through the legs of Courtois to give his side a vital advantage in the tie.
PSG will now look to protect its advantage in the return leg in Madrid on March 9.
Elsewhere, Manchester City all but secured its place in the next round of the Champions League with a dominant 5-0 win against Sporting Lisbon in the first leg.