Rodrygo also scored as Madrid moved back to within four points of leaders Barcelona, who beat Osasuna 2-0 on Saturday.
Madrid had won only two of their last eight games in all competitions and were coming off a 2-1 home loss to Manchester City in the Champions League, prompting jeers from fans at the Bernabeu.Â
Mbappe, who missed the match against City on Wednesday because of a minor injury, scored his 26th goal of the season in all competitions after a good run on the left, passing a defender before firing into the top corner from just inside the area in the 24th minute.
Alaves equalised through Carlos Vicente in a 68th-minute breakaway, but Rodrygo put the visitors ahead again from close range after a run and assist by Vinicius Junior.
The Brazilian hugged Alonso after the goal in a rare show of public support towards the former Madrid player who is in his first season as coach of the club. The gesture came after media reports said the coach had lost the locker room.
Earlier on Sunday, second-to-last Real Oviedo fired coach Luis Carrion after a 4-0 loss at Sevilla.
The result leaves with Oviedo on 10 points from 16 matches, while Sevilla (20 points) moved to ninth place.
Celta Vigo won their first home game of the season in the league by beating seventh-placed Athletic Bilbao 2-0, moving Vigo to eighth and one point behind Athletic.
The game between Socceroos goalkeeper Maty Ryan's struggling Levante and Villarreal was postponed because of heavy rain in the Valencia region.
Villarreal are third and have won six in a row in La Liga. Levante are last, having lost five straight league matches.
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