PSG, who claimed a record-equalling 10th top-flight title last weekend, had to share the points after Anthony Caci's strike gave the hosts a last-gasp equaliser in stoppage time.
Before that, Mbappe had scored his 23rd and 24th league goals. He cancelled out Kevin Gameiro's opener before extending his team's lead following Achraf Hakimi's second-half goal after a Marco Verratti own goal.
The result left PSG on 79 points from 35 games and fifth-placed Strasbourg on 57, two points behind third-placed Stade Rennais who occupy the last Champions League spot.
Adrien Thomasson found the back of the net again seven minutes later, only for his goal to be ruled out for offside.
Mbappe then set up Hakimi for PSG's second goal in the 64th, four minutes before collecting a poor back pass from Alexander Djiku to make it 3-1 for the visitors.
Verratti gave the hosts a lifeline after 75 minutes when he deflected a corner kick into his own net.
Caci then gave Strasbourg a deserved point with a superb half volley two minutes into injury time.