A devastating second-half spell yielded four goals in just eight minutes for the defending Liga champions, who were beaten at Camp Nou by the Basques.
Aleix Vidal's horror injury cast a shadow as Barcelona ran riot in the rehearsal for the Copa del Rey final and gained revenge over Alaves with a 6-0 La Liga victory at Estadio Mendizorrotza.
Luis Enrique's side were beaten on home soil by Alaves in September, but Barca's famed MSN trio helped set the record straight, with Neymar's goal taking him above club legend Ronaldinho in the scoring charts.
All the goals came in two clusters, with Luis Suarez and Neymar striking within three minutes of each other on the stroke of half-time to ruin what had been to that point a resilient display from Mauricio Pellegrino's side, who ended the club's wait for a first Copa final with Wednesday's 1-0 win over Celta Vigo.
Barca's own route to the final proved more problematic, as a heated 1-1 draw against Atletico Madrid at Camp Nou – which sealed a 3-2 aggregate success – came at the cost of red cards for Sergi Roberto and Suarez, though the latter is subject to appeal.
The Uruguayan scored in that match before being given his marching orders and netted twice on Saturday, the first from Vidal's low cross.
Alaves – who had denied Barca a goal on the Catalan giants' last three La Liga trips – had barely recovered from that blow when they conceded a poor second, goalkeeper Fernando Pacheco punching an attempted clearance into Suarez and the loose ball falling for Neymar to tap in.
Christian Santos missed a sitter early in the second half before Barca clicked up a gear with four goals in eight minutes, Messi slotting through Pacheco's legs, Alexis scoring an own goal, Ivan Rakitic spectacularly getting in on the act and Suarez adding his second.
Vidal suffered a nasty looking injury to his right ankle in the closing stages to take the shine off a 15th LaLiga match unbeaten for Barca, who will now turn their attention to Tuesday's Champions League clash with Paris Saint-Germain.
It was Messi who got the first shot away, forcing a smart near-post save from Pacheco inside the opening five minutes.
Samuel Umtiti picked up an early yellow card as he cynically halted a threatening counter-attack from Ruben Sobrino, while a good chance went begging for Vidal – who has recently won a spot in the starting XI – when he blasted over the crossbar on the half-volley.
But it was the hosts who should have taken the lead midway through the half, Atletico loanee Theo Hernandez showing a blistering turn of pace to burst clear on the break, only for Marc-Andre ter Stegen to spread himself and block the full-back's shot.
That missed proved costly as Barca delivered a quick-fire double just before the break, Suarez converting from close range after Vidal supplied the ammunition, with Neymar's 95th goal for the club coming soon after to put the visitors firmly in control.
Santos was left kicking himself after somehow lifting a shot over the bar from just six yards out after Hernandez had sent in a teasing cross from the left.
When Messi pounced on Carlos Vigaray's error and calmly converted, the points were Barca's, but they bolstered their goal difference too with a rampant spell of complete dominance.
Alexis put through his own net under pressure from Messi, while Rakitic's effort was arguably the pick of the bunch as he expertly found the top corner, before Suarez powerfully drove home after Neymar's shot had been saved.
Vidal's graphic injury well and truly took the sting out of proceedings, his ankle turned to an unnatural angle amid widespread concern among his team-mates.
And Alaves, given a clear indication of the size of the task they face in May's Copa final, were left stunned by the rout as Luis Enrique's side moved top ahead of Real Madrid's trip to Osasuna.
