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.
Sociedad had led the league for six consecutive rounds. The team's attractive attack has helped it only lose just once this season.
Madrid's night went from bad to worse when Eden Hazard was forced off on 28 minutes after taking a knock, the latest in a long list of injury setbacks for the Belgium star.
LaLiga: Real Sociedad extend lead at top after win over Cadiz; Valencia play out draw against Alaves
Real Sociedad remained in good form after the international break with a 1-0 win over Cádiz on Sunday for their sixth straight league victory and a three-point lead at the top of the standings.
Karim Benzema, the heart of Madrid’s title run last season, set up Federico Valverde for the 14th-minute opener, and pressured Emerson into scoring an own goal
LaLiga: Valencia register big win in derby against Levante; Real Betis beat Alaves to make winning start
It was a thrilling return to LaLiga for Manuel Pellegrini as Real Betis won 1-0 against Alaves.
Machín takes over following a season in which the Basque Country club finished 16th in the first division of the Spanish league, three points above the relegation zone.
LaLiga: Real Madrid move closer to title after win against Granada; Sevilla secure top four spot after Villarreal loss
Madrid can clinch their first title since 2017 on Thursday with a win at Villarreal, or if Barcelona lose points when they hosts Osasuna.
A statement from Real on Friday confirmed the injury to the Brazilian without stating when he would return although Spanish newspaper Marca said he would be out of action for three weeks.
Zidane had to experiment in Real Madrid's defence with Sergio Ramos and Dani Carvajal serving suspensions and Marcelo and Nacho Fernández injured.
LaLiga: Real Madrid attacker Vinicius Junior to take COVID-19 test again after getting 'wrong' result
Zidane says Vinicius Junior's test was not positive, adding that sometimes the tests come back with an error, instead of a positive or negative outcome.
Valencia have won only one of seven matches since the league resumed and are ninth in the standings, in danger of missing out on European action next season.
LaLiga: Villarreal move closer to Champions League spot after win against Real Betis; Valencia suffer third straight loss
Villarreal's Gerard Moreno converted a seventh-minute penalty kick and added to the lead from inside the area in the 30th to become the league's top-scoring Spaniard with 15 goals
Real Madrid took advantage of rival Barcelona being the first to stumble after the Spanish league resumed, and retook the lead with a 2-1 win at Real Sociedad
LaLiga: Karim Benzema's brace propels Real Madrid to comfortable victory over Valencia; Real Sociedad lose to Alaves
Karim Benzema scored twice and Marco Asensio added another on his return from a 10-month injury layoff as Real Madrid beat Valencia 3-0 on Thursday to keep pace with leader Barcelona in the Spanish league.