Real had squandered two crucial points in what is already a two-horse title race when they had to come from behind to secure a 1-1 draw at bottom side Almeria, the team with the league's worst home record.
Valencia stayed hard on the heels of third-placed Villarreal when Jeremy Mathieu and Pablo Hernandez scored in a 2-0 home win over Deportivo La Coruna.
Barca's victory also set a new points record for the first half of the season, beating the mark of 50 set by Pep Guardiola's side in their treble-winning 2008/09 season.
Malaga's Portuguese midfielder Duda briefly set the home fans' nerves jangling when he curled in a fine free-kick in the 68th minute before Villa netted his second and 14th of the campaign.
"It was a vital win for many reasons," Villa said in a television interview. "There is a long way to go yet although we are of course delighted that we have the advantage at this stage."
In Almeria, Real coach Jose Mourinho gave fit-again Kaka his first start of the season after knee surgery, leaving Karim Benzema on the bench and using La Liga top scorer Cristiano Ronaldo in the striker's role.
The ball broke to Jose Leonardo Ulloa after Pablo Piatti's fine run and the Argentine striker swept it past Iker Casillas from the edge of the penalty area.
"Four points behind under normal conditions is an open league," the Portuguese told a news conference.
"We have performed well in the first half of the season," he added. "We have won 15 games, lost one and drawn only three but that's not enough to win the championship. We hope to improve in the second half."
