Mbappé had his penalty saved by Athletic goalkeeper Julen Agirrezabala in the 68th minute, and Valverde lost possession on defense to allow an easy winner by Gorka Guruzeta in the 80th.
"We knew that he is a good penalty taker," Agirrezabala said of Mbappé. "He missed the last one and I believed that he was going to choose the same side and luckily that's what happened."
Valverde had control of the ball but gave it away while trying to get past a few Athletic players instead of passing it to a teammate, leaving Guruzeta with an easy run into the area for his goal.
Álex Berenguer had put the hosts ahead in the 53td and Jude Bellingham equalised for Madrid in the 78th.
Mbappé had also missed a penalty in Madrid's 2-0 loss at Liverpool in the Champions League last week. He didn't take the one for the club in a Spanish league match on Sunday, but he scored in the 2-0 win over Getafe to ease some of the pressure on him and the club.
But it was another lacklustre outing for Mbappé, who continues to struggle in his first season since finally joining the Spanish powerhouse.
"I won't evaluate the performance of a player because of a missed penalty. Obviously he is sad and disappointed, but you have to move on," Madrid coach Carlo Ancelotti said about Mbappé.
"He is not at his best level, but you have to give him time to adapt. You have to give him time to be able to do better."
The defeat left Madrid four points behind Barcelona, who on Tuesday ended a three-match winless streak in the league with a 5-1 rout at Mallorca.
Madrid, who have a game in hand, had won three consecutive league games since a 4-0 loss at home in the "clasico" against Barcelona.
Madrid has lost five of its last 11 matches in all competitions.
Athletic moved to fourth place with the victory, its fourth consecutive across all competitions.
Both matches on Tuesday and Wednesday were moved forward in the schedule because the clubs will be playing in the Spanish Super Cup in January.
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AP soccer: https://apnews.com/hub/soccer
