In bizarre circumstances, Alaves midfielder Rodrigo Battaglia was shown a straight red card in the 61st minute, possibly for making an offensive remark to the referee or opponent Roberto Lopez while Alaves were about to take a throw-in.
Sociedad had been dominating the Basque derby before then and increased the pressure once they had the man advantage, but Alaves comfortably repelled their many crosses into the box.
Alaves keeper Fernando Pacheco was also on top form and made five saves during the game, the best of which came when he flew to his right to tip away a vicious strike from Willian Jose in the second half.
Sociedad had started the weekend top of the table but the draw left them second in the standings on 25 points after 12 games.
Leaders Atletico, who beat Real Valladolid 2-0 on Saturday, have 26 points after 10 matches.
Sociedad have drawn their last four games, having been held 1-1 by Villarreal last Sunday while failing to beat AZ Alkmaar and Rijeka in their last two Europa League outings. Before the draw with Villarreal, they had won their previous six league games.
Next up for Sociedad is Thursday's Europa League group stage decider away to Napoli, which they need to win to guarantee their place in the knockout stages.
"I think the team did enough today to get the three points, but lately we have been unable to take our chances to win games. We have to keep trying," said Sociedad coach Imanol Alguacil.
"The hardest thing in football is creating chances and this team creates many, many chances, but once again our opponents' best player was their goalkeeper."
The visitors were hoping to benefit from the return of David Silva from injury but the playmaker hurt a muscle in the warm up, dealing another blow to the team after forward Mikel Oyarzabal, the league's top scorer, was injured against Rijeka.
Alguacil said he expected Silva to return for the trip to Naples but admitted Oyarzabal was unlikely to make it.
The coach also said the team were feeling frustrated after the result but still believed they could win against Napoli.
He added: "Right now our spirits are a little low because we didn't win but we've played well enough to win every game this season and we'll go to Napoli with that same aim. No-one in the team has any doubt we are capable of winning there."