Griezmann opened the scoring in the 12th minute while Messi struck twice before the break to make it four goals in his last two matches and go top of the league's scoring charts with 11 goals.
French forward Griezmann scored a classy second goal after halftime while things got worse for Granada when Jesus Vallejo was sent off in the 78th minute for clattering into Barca substitute Martin Braithwaite.
The victory took Barca back into third place in the standings with 34 points, closing the gap with leaders Atletico Madrid to four points.
Barca had produced one of their best displays under Ronald Koeman in Wednesday's 3-2 win at Athletic Bilbao but have tended to struggle for consistency this campaign, in particular away from home.
But Griezmann got them off to a strong start when he stuck out his leg to prod the ball home when a deflection from Granada's Roberto Soldado fell into his path and played him onside.
The Argentine struck again shortly before halftime, arrowing low into the net from a freekick and then enjoyed a rare rest when Koeman removed him for Braithwaite midway through the second half.
Griezmann showed his soaring confidence with his brilliant second goal, controlling a cross from Ousmane Dembele with his left foot then using his other foot to find the far bottom corner from a tight angle with his right.