PSG were on a high having claimed a record-equalling 10th top-flight title last weekend but were quickly brought back to reality 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 strike.
The result left PSG on 79 points from 35 matches and fifth-placed Strasbourg on 57, two points behind third-placed Stade Rennais who occupy the last Champions League spot.
“We are not relaxing, the season is not over,” PSG coach Mauricio Pochettino said. “We have the desire to play until the end of the season.”
The hosts went ahead after just three minutes when Gameiro outpaced Presnel Kimpembe and beat Gianluigi Donnarumma with a low shot.
In the 23rd minute, PSG levelled when Neymar found Mbappe with a delightful through ball, and the France forward finished with a low shot that went between the legs of Matz Sels.
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.
Marco 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.
Meanwhile, Tottenham manager Antonio Conte has slammed "fake news" reports that he is a contender to take charge at PSG.
Conte was linked with the French champions this week after reports emerged that Pochettino is likely to leave at the end of the season.
The Italian has 12 months left on his Tottenham contract and has refused to commit his long-term future to the Premier League club since replacing the sacked Nuno Espirito Santo in November.
Asked about the link to PSG, Conte insisted there was no truth to the claims.
"Obviously I think that it's good that other clubs appreciate my work. This is one thing. But the truth is that I don't like when people try to [invent] news - only to speak, only to create problems," Conte told reporters.
"This is not right, not fair for me, for the clubs involved and for my players. I repeat, this type of situation makes me smile.
"But I think that also the people that want to tell something about this have to show respect for all the people involved in this situation, and not invent fake news and tell a lot of lies.
"In this situation, who works in football - players or managers - know very well that at this point of the season, a lot of fake news starts to go out."