Mbappe struck in each half for Madrid, including the winner in the 87th minute. Raphinha scored for the Catalan club in the second half.
The result on Sunday (Monday AEDT) ended Barcelona's bid to become the first team to win their opening 10 La Liga matches. They had matched Real Madrid's nine victories from nine games in 1968/69.
Madrid stayed in third place but moved within three points of Barcelona. Atletico Madrid are second, two points behind Barcelona.
Earlier, Ballon d'Or contender Harry Kane failed to find the net, but Bayern Munich defeated Freiburg 3-1 on the road to regain first place in the Bundesliga.
Their goals instead came from Dayot Upamecano, Serge Gnabry — his 100th in the Bundesliga — and Michael Olise.
Bayern are tied on 17 points with Borussia Dortmund, who beat Eintracht Frankfurt 1-0 on Saturday.
Bayern remained without coach Vincent Kompany, who collapsed during training on Friday.
Freiburg had won eight of their first nine matches in all competitions, but now have lost three straight, and remain third in the Bundesliga.
In France, US international Tim Weah scored Marseille's second goal as they beat Le Havre 4-3 to move to eighth place.
The hosts opened a four-goal lead by the 55th minute and held on after conceding in the 77th, 86th and 89th.
Rennes beat Strasbourg 3-0 with a pair of goals by Sebastian Szymanski to take first place with 16 points, one more than both Monaco and Lille, who drew 1-1. Lille played with 10 men from the 66th after a red card for Tanguy Nianzou.
Paris Saint-Germain's home game against Lyon was moved from Sunday to Monday at the request of Paris police amid mass protests in France. PSG can move back to first place with a win.
Weah's US teammate Christian Pulisic scored six minutes into Milan's 2-0 win at Udinese in Serie A.
It was Pulisic's second goal this season with Milan, who moved back to fourth place. Milan were coming off a 3-0 loss at Bournemouth in the Europa League.
Earlier, Marcus Thuram scored and then earned the penalty kick that decided Inter's 2-1 win over visiting Fiorentina.
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