Rodrygo also scored as Madrid moved back within four points of leader Barcelona, which beat Osasuna 2-0 on Saturday.
Madrid had won only two of its last eight games in all competitions and was coming off a 2-1 home loss to Manchester City in the Champions League, prompting jeers from fans at the Santiago Bernabeu Stadium.
“We are together in this, we keep fighting in the good and the bad moments,” Alonso said. “The team competed well today. We started well and came away with the three points. It's fundamental to be united, it's the only way to remain solid and improve day-to-day.”
Alaves equalized through Carlos Vicente in a 68th-minute breakaway, but Rodrygo put the visitors ahead again from close range after a run and assist by VinĂcius JĂşnior.
It was Rodrygo’s second goal of the season. He had scored for Madrid against City to end a nine-month scoring drought with the club.
VinĂcius JĂşnior hugged Alonso after the goal in a rare show of public support by the Brazilian toward the former Madrid player. The gesture came after media reports said the coach had lost the locker room.
Alaves, which stayed in 12th place, was coming off a win against Sociedad.
Goalkeeper Thibaut Courtois earned his 200th victory in 310 appearances with Madrid since he arrived in 2018. Defender Antonio RĂĽdiger played in his 100th Spanish league match with the club.
Alonso used 19-year-old VĂctor Valdepeñas, from Madrid's “B” team, in the left back position because of injuries to the regular starters.
Oviedo coach fired
Second-to-last Oviedo fired coach Luis CarriĂłn after a 4-0 loss at Sevilla.
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The result extended Oviedo's winless streak to 10 matches in all competitions. Oviedo hasn't won since a 2-1 victory at Valencia in the Spanish league in September.
CarriĂłn leaves with Oviedo on 10 points from 16 matches.
Sevilla moved to ninth place with 20 points.
Celta's home win
Celta Vigo won its first home game of the season in the league by beating seventh-placed Athletic Bilbao 2-0.
The result moved the Vigo club to eighth place, one point behind Athletic.
Nico Williams missed a late penalty kick for Athletic.
Levante-Villarreal postponed
The game between Levante and Villarreal originally scheduled for Sunday was postponed because of heavy rain in the Valencia region. The league said it made the decision after safety recommendations made by local authorities amid a weather alert issued in the region.
Villarreal was in third place and has won six in a row in the league. Levante was last and has lost five straight league matches.
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