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."
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."
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.
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.
