Negredo put Valencia on their way against visiting Granada with a goal in the seventh minute, and he converted two second-half penalties after Rodrigo Moreno had doubled the advantage.
Valencia are still without a win in four La Liga matches under Neville, and had previously only beaten third-division team Barakaldo in the cup since the former Manchester United defender took charge.
Negredo has been given a new lease of life under coach Neville and the win was a confidence boost for Valencia.
He robbed the ball off Diego Mainz and stroked it into the corner past keeper Ivan Kelava for his first goal.
Santi Mina laid the second on a plate for Rodrigo and then Mina was fouled by Uche Agbo with Negredo scoring the subsequent penalty after 62 minutes.
Negredo, the former Manchester City player who has had appendicitis and was frozen out by former-coach Nuno Espirito Santo, scored his third from the spot in the 83rd minute after he was fouled by Matheus Doria.
A confident Villarreal had won their previous five matches and went ahead through Leo Baptistao and Samu Garcia but Bilbao came back fighting after the break.
Baptistao slotted home clinically following a good team move after 16 minutes and Samu Castillejo then sent a defence splitting pass for Garcia, who rounded keeper Iago Herrerin and knocked the ball into an empty net.
Williams led the comeback with a 53rd minute strike, after being set up by Benat Etxebarria, and then found Aduriz to score from the centre of the penalty area.
Laporte completed the turnaround by converting a Benat free-kick from a tight angle after 80 minutes.
A poor second half for Villarreal also saw the dismissal of Mario Gaspar with three minutes to go.