Madrid, meanwhile, had to fight back from 2-0 down with 10 men to rescue a point in a 2-2 draw at lowly Osasuna, in a game which saw Gareth Bale taken off yesterday.
Following on from his first Barca hat-trick in the 6-1 demolition of Celtic in midweek, Neymar made the difference once more as he converted from the spot after Mario Gaspar had blocked Jordi Alba's cross with his hand.
Villarreal were level just three minutes after the interval when Mario Musacchio bulleted home a Manu Trigueros corner.
"Apart from the game against Atletico Madrid in the Super Cup, where we won a trophy, this is the game that has given me most satisfaction since I took over," said Barca boss Gerardo Martino.
"We have achieved a level of consistency in the last three games against Cartagena, Celtic and today that is pleasing.
"It was a great win for the manner in which we played and the quality of opponent we were facing rather than because Madid had dropped two points."
And it took the visitors just three minutes after the restart to level as Musacchio took advantage of some awful marking to head high past Jose Manuel Pinto.
But Villarreal goalkeeper Sergio Asenjo was left helpless when Sanchez latched onto Cesc Fabregas's pass behind the away defence and squared for Neymar to finish from close range.
Earlier, Madrid looked set for their third league defeat of the season after two headed goals from Oriol Riera put Osasuna 2-0 ahead inside 39 minutes.
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