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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.
The victory allowed second-placed Real to stay in touch with leaders Barcelona, who also have 28 points after 13 games but top the standings due to their superior goal difference.
The win means Madrid climb up to second in Group A, but five points behind leaders Paris St Germain.
Barcelona sailed back to the top of the table and trouble returned to Real Madrid after they endured their first La Liga defeat of the season at newly-promoted Real Mallorca.
Real Madrid are top with 15 points from seven matches and Atletico stayed third with one point less.
Real Madrid climbed to the top of the standings with 14 points from six games, one point and one place ahead of Atletico Madrid, whom they face on Saturday.
Here are 10 must-read stories from around the globe this Sunday.
Real, European champions 13 times and three years in a row from 2016-18, failed to muster a shot on target and suffered their first defeat of the season.