Barca struggled to find their rhythm at the Nuevo Los Carmenes stadium on a poor pitch but took the lead in the 25th minute when Ivan Rakitic smashed the ball in from close range after Granada defender Cala failed to clear a Suarez centre.
Lionel Messi twice went close before Suarez ran on to a chipped Rakitic pass and slipped the ball past goalkeeper Oier to make it 2-0 three minutes into the second half.
Barca appeared to be cruising against their second-bottom opponents but Granada pulled a goal back five minutes later when Marc Bartra felled Lassane Bangoura in the penalty area and Fran Rico stroked the spot kick past Claudio Bravo.
Suarez then sped clear in the 70th minute, left Oier floundering and unselfishly set up Messi to find the empty net.
The win capped another fine performance from the Uruguay international, who struck twice in Tuesday's 2-1 Champions League victory at Manchester City after Barca had been beaten 1-0 at home by Malaga last weekend.
Messi fluffed two chances in the closing stages when sent clear by Neymar, leaving the Argentina captain on 27 league goals this season, two behind top scorer Cristiano Ronaldo of Real Madrid.
"I think it was important for us to keep racking up points after a tough game against Malaga," said Rakitic. "It's always a difficult challenge here but I think we worked really hard and deserved the win."
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