Uruguay forward Suarez scored his 100th Barca goal in Wednesday's 3-1 Copa del Rey victory over Athletic Bilbao and he broke the deadlock here during a first half dominated by the home side at Camp Nou.
Andre Gomes - who had one of his best games in a Barca shirt - and Rafinha were among those to be denied in the opening period, but Messi - who also scored against Athletic - capitalised on goalkeeper Javi Varas' error to make it 2-0 in the 52nd minute.
Suarez grabbed his second with a cool finish five minutes later before Arda Turan followed up on the rebound from the Uruguayan's saved effort to make it three goals in seven second-half minutes.
That put Barca out of sight, before Aleix Vidal completed the rout with 10 minutes left with his first goal for the club.
Luis Enrique's men consequently rebounded from last week's contentious 1-1 draw at Villarreal to close within two points of leaders and rivals Real Madrid, albeit having played two games more.
Zinedine Zidane's side travel to Sevilla - who drop to third below Barca - on Sunday.
Barca were typically dominant in possession in the early stages without causing much trouble in the opposition's third.
Las Palmas were caught cold in the 14th minute, though. Gomes sprung the offside trap on the right before squaring to Suarez, who swept the ball home first time from eight yards.
That proved the catalyst for a Barca onslaught and Rafinha soon dragged a shot wide of the right-hand post after fine work by Messi, with a frustrated Suarez wide open.
They were in full flow and a terrific break involving Arda and Suarez led to Gomes drawing a terrific point-blank stop from Varas.
Gomes then controlled Sergio Busquets' excellent chipped pass only to lash a half-volley over, before Messi pounced on Varas' spill to flash a cross-shot across the face of goal after again showing sublime footwork.
Messi almost scored a fourth free-kick of 2017, but Varas palmed his effort away from the top-right corner before the goalkeeper made a fine one-on-one save to deny the Argentinian superstar.
Barca rested captain Gerard Pique, perhaps in part due to the defender's public criticism of officials following matches against Bilbao and Villarreal, but the visitors did not threaten until Kevin Prince-Boateng's attempted overhead kick early in the second half.
