Suarez opened the scoring on 14 minutes following a low cross from Andre Gomes.
Messi tapped home to make it 2-0 early in the second half after Las Palmas goalkeeper Javi Varas failed to handle a left-flank cross by Rafinha and left the ball in front of the goal.
It was Messi's 13th goal in the last 12 matches, and his first against Las Palmas, the only Primera Division club yet to concede a goal to the playmaker.
Suarez added his second with a well-placed shot into the far corner, and two minutes later set up Turan after his own shot was saved. Vidal completed the rout with 10 minutes to play.
An early goal from the Argentina international after just eight minutes proved enough at Vicente Calderon. Diego Simeone's side moved two points behind Sevilla as a result.
In the day's first match, Leganes shared a goalless draw with Athletic Bilbao.
Leganes came closest to scoring but forward Darwin Machis squandered a few great opportunities, including one-on-one chances in front of the goal.
Athletic had a goal by Raul Garcia ruled out for a foul in the buildup.
It was Athletic's second straight goalless draw against a newly promoted side following last week's stalemate against Alaves.
Saturday's late match saw Deportivo La Coruna held to a goalless draw by Villarreal at Estadio Riazor.
Information from The Associated Press was used in this report.