The Brazilian rolled in the opener after a pass from Luis Suarez, before repaying the favour seconds later for the Uruguayan to clip in a second.
Las Palmas were given hope when defender Pedro Bigas swept in Kevin-Prince Boateng’s low cross.
But two more quick-fire goals from Neymar killed off the home side.
Neymar stooped to head in Ivan Rakitic’s deep right-wing cross and then squeezed in Jordi Alba’s pass for his first hat-trick in an away game for the Catalan club.
Barca are level on points with Real Madrid, who thrashed Sevilla 4-1 at the Santiago Bernabeu, but although they have the superior head-to-head record they have played one game more.
Madrid will win their first La Liga title since 2012 if they win both of their remaining games.
Zinedine Zidane’s side play their game in hand when they go to Celta Vigo on Wednesday, ahead of a final-day trip to Malaga next weekend.
Barcelona, who will play their final game under departing boss Luis Enrique, host eighth-placed Eibar at the Nou Camp next Sunday.
