Suarez finally levelled for the Catalans in the 72nd minute, scoring for the eighth league game in a row after some impressive saves from visiting goalkeeper Fernando Pacheco had kept Barca at bay, while Messi's sumptuous free kick in the 84th gave them a seventh consecutive league win.
Suarez volleyed home after being teed up by Andres Iniesta for Barca's equaliser before Messi's brilliant set-piece near the end sealed the points. Lionel Messi then duly stepped up to put a free-kick in the top corner after 84 minutes to ensure a hard-fought win for the Primera Liga leaders.
Naturally, it got social media talking.
Playing in an unfamiliar right-wing position in this match, Coutinho struggled to reach the heights of that debut though, and indeed the heights of his career at Liverpool that endeared him to the Anfield faithful.
Barcelona's attacking talent just could not click and Alaves passed up a good chance to double their lead as Ruben Duarte saw his shot blocked following good work from Guidetti.
They also pushed Valencia close in the Copa del Rey quarter-finals before losing on penalties in midweek, and Valverde said: "They are in their best form of the season".
The visitors continued to push and look for a second goal, as they knew that no matter how hard they tried at the back, Barcelona's quality would inevitably shine through.
Messi is said to have urged Valverde to replace Nelson Semedo and Lucas Digne with Sergi Roberto and Jordi Alba.
But while Suarez will want the focus to be on his role in helping Barcelona turn the match around, it is more likely to land on an incident in the 86th minute.
It was a classy performance from start to finish by the hosts.
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