Messi set up Neymar to give the visitors the lead before missing a first-half penalty after he was fouled by Loukas Vyntra, who was sent off.
Barcelona doubled their lead through Ivan Rakitic, before Messi’s pass allowed Sandro Ramirez to make it 3-0.
Pedro made it 4-0 and Messi added a fifth as Barcelona went top of La Liga.
Sevilla had earlier leapfrogged Barca after a 3-1 win at Cordoba.
But the 22-time champions of Spain went two points clear at the summit after their biggest win since Enrique took charge in the summer.
The Catalans have won all four league games, scoring 11 times, and have yet to concede.
They also edged past Apoel Nicosia 1-0 in the Champions Leaguegroup stage on 17 September.
This win was every bit as comfortable as the scoreline suggests, although there was one moment of concern when Neymar landed awkwardly after a challenge.
The Brazil forward eventually climbed to his feet before being helped off the pitch with what appeared to be an ankle problem.
However, the injury is not thought to be serious, with Neymar clearly disappointed that he was not allowed to continue.
Barcelona are next in action on Wednesday when they travel to mid-table Malaga, while Sevilla entertain Real Sociedad on the same evening.