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The draw left champions Barca top on 36 points after 17 games with Real also on 36, five clear of third-placed Sevilla.
The victory means Zinedine Zidane's side move to the top of the table, for a few hours at least, before Barcelona play later on Saturday at home to Real Mallorca.
Zinedine Zidane's Real side demonstrated their new-found grit to re-take the lead four minutes later.
PSG pulled off an unlikely turnaround as Mbappe capitalised on a defensive mix-up to pull one back in the 81st minute before Pablo Sarabia, once of Real, drove into the top corner second later.
Portugal captain Cristiano Ronaldo scored another treble, moving closer to 100 international goals, as he led the European champions to a 6-0 drubbing of Lithuania in a qualifier.
Earlier, Madrid were knocked out of the Champions League at the last-16 stage by Ajax Amsterdam.
The last-gasp win left Madrid third in the standings with 57 points after 29 games, while second-placed Atletico Madrid have 59 and runaway leaders Barcelona are on 69.
Ajax, who have won the trophy four times, most recently in 1995, reached the last eight of Europe's top competition for the first time since 2003.
A clinical Barcelona won 3-0 at Real Madrid in an intense second leg of their Copa del Rey semifinal as a Luis Suarez double helped them secure a 4-1 aggregate victory and a spot in the final for the sixth straight season.
A clinical Barcelona won 3-0 at Real Madrid in an intense second leg of their Copa del Rey semi-final on Wednesday.