Tino Costa and Nelson Valdez both found the net inside the final two minutes to deny Bilbao, who had lost Ander Herrera to a needless red card midway through the second half, a first away win of the season.
Valencia started the game on the front foot and went close to breaking the deadlock when Soldado rattled the post in the fifth minute.
The opening goal came against the run of play on 19 minutes, Aduriz capitalising on a misplaced clearance and placing his shot beyond Vicente Guaita from 20 yards.
Aduriz put his side back in front with an instinctive close-range finish on the half-hour.
Then game turned dramatically on 67 minutes when, for no apparent reason, Herrera kicked out at Rami and was rightfully sent off.
Valencia took control of the match thereafter and bombarded Iraizoz's goal with shots from all angles.
The experienced stopper did well to deny fierce efforts from Soldado, Costa and Antonio Barragan.
However, he was beaten for the second time on the night when Costa slid home a rebound from eight yards after Valdez's initial shot had been cleared off the line.
The Valencia pair switched roles to seal the win in stoppage time, as Costa swung in a cross and Valdez headed home.
Valencia are now eighth in the La Liga table, having risen six places.