Crosstown rivals Atletico Madrid earned their first win of the season before facing Liverpool in the Champions League, while Athletic Bilbao lost before hosting Arsenal.
France striker Mbappe opened the scoring in the 12th minute with his fourth goal in as many games.
Xabi Alonso's Madrid played a man down for nearly an hour after new central defender Dean Huijsen received a red card in the 32nd for pulling down Mikel Oyarzabal with only the goalkeeper to beat.
Mbappe helped double the lead in the 44th when he used some fancy dribbling to spin around his marker near the byline before finding Arda Guler with a pass for the Turkey midfielder to score.
Oyarzabal pulled one back for the hosts by converting a penalty in the 56th.
"The red card changed the game a bit. We struggled a bit as a team, but we take home the three points," Mbappe said after Madrid's fourth win in as many games to start La Liga.
"Our start to the season has been very good, as has mine."
In the capital, Nico Gonzalez scored on his debut for Atletico after the Argentina midfielder joined from Juventus on loan to help beat Villarreal 2-0.
Pablo Barrio netted in the ninth minute before Gonzalez headed in Marcos Llorente's cross in the 52nd.
Atletico's first win of the season comes after their worst start since Diego Simeone became coach over a decade ago.
Elsewhere, Alaves won at Athletic Bilbao for the first time in 20 years thanks to an own goal by Alex Berenguer to take the Basque Country derby 1-0.
It was the first loss for Athletic this season after Ernesto Valverde's side had won their first three matches, while Getafe continued to impress with a 2-0 win over promoted Real Oviedo for their third victory in four.
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