The Catalans' hopes of winning a third straight title rest on Real Madrid losing one of their remaining games.
With his first goal, the Argentine became the first player in Europe's top five leagues to score 50 goals this season.
Barca's formidable front three of Lionel Messi, Luis Suarez and Neymar all scored as the champions inflicted the heaviest league defeat of the season on Villarreal, who have the second best defensive record in Spain's top flight.
Barcelona holds the tiebreaker over Madrid, but Madrid has three matches to play after this round to Barcelona's two.
Messi saw a free kick easily saved by Villarreal keeper Andres Fernandez, but Barca's persistence paid off in the 21st minute when Neymar was able to open the scoring.
Neymar, meanwhile, now has nine goals in nine games against Villarreal after opening the scoring at Camp Nou.
"We have to win every match until the end to keep putting pressure on Madrid", Barcelona forward Luis Suarez said.
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And 13 minutes, Jaume Costa handled the ball in the Villarreal box to gift Lionel Messi a penalty and his second goal of the day. However, Real Madrid still own the destiny of the title as they have a game in hand to play against Celta Vigo.
James Rodriguez took the game away from the hosts in just 11 minutes as he scored in what might be his easiest brace in his career.
Diario Sport previously reported that the offer to Messi will be until 2022 and that recent new deals for strike partners Luis Suarez and Neymar have laid the foundations for him to commit his future to the club.
Atletico guaranteed a top-four finish as Saul Niguez's victor 21 minutes from time earned a 1-0 win over Eibar.
They have not lost to the Yellow Submarine in their last 16 La Liga meetings, stretching back to March 2008.
Despite struggling to draw at El Madrigal in January, Barca will emulate their 3-0 victory over Villarreal last season, with a team devoid of European duties at this stage finding extra energy to dominate the Europa League hopefuls.
But Enrique is focused exclusively on securing three points against a Villarreal side who are hoping to keep their European ambitions alive following three successive league victories, including a 1-0 win at Champions League semi-finalists Atletico Madrid.