Alcacer was immediately substituted for Gerard Deulofeu after his second goal, but Barca were back on the attack soon after and Suarez wanted handball against Clement Lenglet when his shot struck the Sevilla defender's arm from close range, the striker having been found by Messi.
Alcacer got in front of his marker to prod home a brilliant Ivan Rakitic cross and seal another three points for Valverde's men, who now have 34 from a possible 36 in La Liga.
Messi attracted a little extra attention by becoming the third Barcelona player to reach the 600-match milestone, behind captain Andres Iniesta (643) and Xavi Hernandez (767).
Luis Muriel showed that Samuel Umtiti's not quite the complete player yet by robbing him of the ball in the first half and firing over Marc-Andre ter Stegen's goal.
Sky Go is also available on iOS and Android smartphone and tablet devices as well as other platforms including Amazon Fire devices, iPad, Mac computers and laptops, Playstation 3, PS4 and Xbox One. With the form Messi is in, it would almost be a surprise if he didn't add to that total today night.
There was no stopping Pizarro in the 59th minute, though, as he attacked an Ever Banega corner ahead of Pique and directed a terrific downward header into the far corner.
Before the match, banners were unveiled either side of a huge national flag (the Senyera) with a message in Catalan and English: "Justice".
Busquets 7: Missed an early close-range chance created by Messi but was otherwise impeccable, using the ball superbly and showing great positional intelligence to knit the team together.More news: Xbox One X review: A console that keeps up with gaming PCs
"First of all, you have to try to win, but we do so by dominating, adopting a high line and recovering balls".
Although all eyes were on Lionel Messi playing in his 600th game for Barca, it Alcacer who was on target for his side in the pouring rain in the Catalan capital.
Atletico's second win in nine matches ended a streak of three consecutive 1-1 draws in the league and Copa del Rey.
"We know how difficult it will be to keep this up, it's practically impossible".
"Paco had a very good game, but with all the defensive work he'd put in, he was tired", Valverde explained.
Pressure still mounts on the Barca manager, as they drew 0-0 with Olympiacos on TuesdayWhat time is kick-off? That's the first three minutes covered.
However, their efforts came to no avail, and referee José González González's whistle following four additional minutes signalled Barca reclaiming their four-point lead at the top of La Liga heading into the upcoming international break.
"He got two important goals and we hope that they give him the confidence that strikers need", Valverde told a press conference.
Ernesto Valverde says he's more worried with results than winning with style. Struggled to get involved down Sevilla's right flank, coming out very much second best in his personal duel with Alba.