A 1-0 defeat at home to the leader coupled with Lille's 2-0 win over Metz left Mauricio Pochettino's men six points adrift heading into the match at Stade de la Meinau.
But Mbappe backed up a double in PSG's fine 3-2 first-leg triumph at Bayern Munich in their UEFA Champions League quarter-final tie with his 21st Ligue 1 goal of the season.
That preceded efforts from Pablo Sarabia – his third in four games – and Moise Kean before the break, with Dion Moise Sahi replying for the hosts prior to Leandro Paredes executing a fine free-kick to wrap up the points.
For all its domination of the ball, PSG was almost caught cold when Adrien Thomasson's 30-yard thunderbolt cannoned off the left post.
But PSG was ahead after 16 minutes when Mbappe showed Lamine Kone a clean pair of heels and drilled a left-foot shot through Matz Sels's legs from a tight angle.
The lead was doubled by Sarabia, who turned brilliantly after collecting Danilo Pereira's pass, shimmied past Kone and rolled coolly past Sels.
Kean made sure of the points before the break with a clinical finish from Mbappe's poked throughball.
The Everton loanee forced Sels into a save on the hour but it was Strasbourg who scored next, Sahi forcing his shot pass Sergio Rico – on for Keylor Navas at half-time – from Jean-Ricner Bellegarde's right-wing cross just 73 seconds after entering the fray.
Sanjin Prcic curled one just wide of the left post for Strasbourg, but Paredes picked out the right side of the goal from 30 yards to complete the win from a set-piece won by another marauding Mbappe run.