It seemed Valencia would suffer a setback to their hopes of making a run for a European place next season when Granada took the lead early through Piti in the second half after a blunder from Javi Fuego.
Valencia got level thanks to substitute Paco Alcacer, but the visitors clung to 1-1 heading into stoppage time.
That was when 19-year-old Vezo intervened, meeting a free-kick from the right at the far post and keeping his composure to cushion the ball home.
Valencia have found themselves mired in mid-table but have higher aims.
They showed their quality on a trip to Cyprus, where they beat Dynamo Kiev 2-0 in a relocated Europa League clash on Thursday.
In Sunday's return to Primera Division action, Valencia felt they should have had an early penalty when the ball appeared to strike Brayan Angulo's arm inside the area, but they were denied.
The lively Juan Bernat then created a chance for Sofiane Feghouli with a cross from the left but the Valencia striker could not make the required contact.
Granada had a half-chance of their own in the 39th minute when Piti made excellent ground on the left and fed a low lay-off to substitute Mohammed Fatau by the edge of the penalty area.
Fatau had a clear sight of goal but sliced his low shot wastefully wide.
Shabby defending by Fuego gifted Granada the opening that resulted in the 47th-minute opening goal.
He chested down a cross from the right delivered by Allan Nyom, but instead of finding a team-mate it sat up perfectly for Piti to drill a low bouncing shot across goal and into the left corner.
Alcacer was soon introduced off the bench and his impact was almost immediate.
The leveller came after 64 minutes when Feghouli crossed from the right and Alcacer beat his marker to the ball at the near post, clipping a neat finish wide of the goalkeeper.
It was a tight offside call, with Alcacer making his run off the last defender, but the assistant referee kept his flag down.
Valencia had the upper hand but a second goal for the home side looked like remaining elusive until Vezo, the former Vitoria Setubal player, had his say.
Michael Pereira took down Dani Parejo on the flank, and the Valencia player got up to take the free-kick himself, with Granada goalkeeper Roberto gambling and losing by coming to meet the ball.
He appeared to get there first but flicked it across goal to Vezo, who had an empty net to aim at.