The lanky striker crowned a towering performance with a clever lob on the counter in the fourth minute of stoppage time, as Valencia piled forward to try to grab an equaliser.
With five games left, Sociedad moved on to 58 points, five clear of Malaga and Valencia in fifth and sixth respectively. Malaga beat Getafe 2-1 earlier in the day.
This season's surprise package Sociedad closed to within 10 points of third-placed Atletico Madrid, who fell 2-1 at home to second-placed Real Madrid on Saturday.
"It's a big step forward for us, but it isn't definitive," Agirretxe told Spanish broadcaster Canal Plus.
"More than feeling pressure we are enjoying ourselves and it is better than fighting to avoid relegation."
Valencia took the game to the hosts and Spain striker Roberto Soldado put them in front after 25 minutes, getting in front of his marker to glance a superb header inside the far post.
Sociedad's Chilean keeper Claudio Bravo was lucky to still be on the pitch soon after as the match officials failed to spot him handling the ball well outside his area as he blocked Soldado's attempted lob.
"It's a big decision because with the score at 1-0 and them down to 10 men it could have made a big difference," Soldado said.
"But the referee didn't see it, and in the end they have been better."
Sociedad hauled themselves back into the game when Inigo Martinez hooked in a volley at the back post after a well-worked corner move in the 34th.
SEARING COUNTER-ATTACKS
Two searing counter-attacks put Sociedad in charge, the first coming in the 73rd minute when Agirretxe and Gonzalo Castro tore down the pitch interchanging passes, and Uruguayan Castro stroked in the finish.
Agirretxe volleyed in the third at the back post in the 86th and settled nerves in stoppage time after Jonas had nodded one back for the visitors.
Malaga kept up their push for a possible return to the Champions League when Roque Santa Cruz headed them in front just before halftime against Getafe.
Defender Weligton doubled the lead with another header, this time from a corner, two minutes after the restart.
Getafe pulled one back with a close-range volley from Juan Valera in the 70th minute and Manuel Pellegrini's side endured some nervy moments near the end as they missed a number of chances to settle the game.
Malaga may not be able to take up a place in European competition next year if they are successful as they have been handed a one-season ban by UEFA due to financial irregularities.
The Costa del Sol club have appealed to the Court of Arbitration for Sport (CAS) and a decision is expected in mid-June.
Sevilla's fading hopes of qualifying for European competition were not helped by a 1-1 draw at Real Valladolid when Alvaro Negredo equalised for them from the penalty spot.
