Sevilla gave Real Madrid the chance to move eight points clear at the top of La Liga after they were held to a goalless draw.
But the result flattered Madrid, who struggled to break down Alaves until Asensio unleashed from distance at the Santiago Bernabeu, where the disgruntled fans had whistled with the score goalless at half-time.
He received treatment on the ground for several minutes and a stretcher was brought out, although the Argentinian was able to walk off the pitch.
A superb second-half strike from Memphis Depay was quickly cancelled out by an equally impressive run and finish from Alaves' Luis Rioja.
Manchester United manager Ole Gunnar Solskjaer is fighting to keep his job heading into an away match at Tottenham
The Gunners are languishing in 13th place with 10 points despite the run. Arsenal are immediately above Palace who have 10 points.
LaLiga: Atletico Madrid fail to go on top of league after being stunned by last-placed Deportivo Alaves
Victor Laguardia scored with a header off a corner four minutes into the match, giving Alaves their first points of the season and preventing Atletico from regaining the league lead.
The games between Barcelona and Sevilla and Villarreal and Alavés, originally scheduled for Saturday, will be played at later dates. They have not been rescheduled yet.
Friday's win left Valencia top of the standings with seven points from three matches, along with Mallorca, ahead of the rest of the round.
Sevilla finished with 77 points. That is one more than the previous record set by the team in 2014-15.
LaLiga: Diego Simeone's Atletico Madrid on cusp of league glory but Real Madrid determined to pounce
Real Madrid will have to win at home against Villarreal to have any chance but the omens looks good for Zinedine Zidane's side, given their opponents may not be at full strength.
Jan Oblak's 86th-minute heroics ensured Luis Suarez's 500th career goal was enough for a 1-0 victory at the Wanda Metropolitano, meaning the gap at the top of LaLiga remains at four points, with 10 games left to play.
European football weekend: Barcelona chase Atletico Madrid, French showdown and prolific strikers duel
Atletico Madrid look to keep Barcelona at arm's length in Spain and there is a huge showdown at the top of the table in France this weekend in Europe.
LaLiga: Deportivo Alaves skipper Manu Garcia says his club is 'privileged' to stay in top-flight during centenary celebrations
Alaves, who were founded on 23 January 1921, have not had much success over the years having spent most of the seasons outside the top-flight. Their most memorable achievement to date remains reaching the UEFA Cup final in 2000-01, where they narrowly lost to eventual champions Liverpool 5-4 after extra time.
Atletico Madrid's win was a significant psychological boost on the back of slipping up for the first time in nine games at home to Celta Vigo earlier this week.
Mato redirected a low cross by Martin Aguirregabiria just inside the far post for the 66th-minute winner.
Zidane's position has come under scrutiny again after Wednesday's defeat by third-tier side Alcoyano in the Copa del Rey but Madrid responded at Mendizorroza without their coach, or their captain Sergio Ramos, who was out with a knee injury.
European football matchday: Real Madrid to play Alavés without Zinedine Zidane; Manchester City vs Cheltenham in FA Cup,
A look at what’s happening in European football on Saturday
Valencia ended an eight-game winless streak and moved further from the relegation zone with a 1-0 win at Valladolid in the Spanish league on Sunday.
LaLiga: Barcelona beat Huesca in Lionel Messi's 500th game; Atletico Madrid back on top after 1-0 win over Alaves
Although Ronald Koeman's side began a potentially season-defining run of four consecutive away matches with a comfortable win, they remain a distant 10 points behind Atletico in fifth having played a game more.