Visiting goalkeeper Tono pulled off a string of magnificent saves to keep Barca at bay while his opposite number Victor Valdes saved a clear one-on-one before Xavi thundered a shot in off the underside of the bar.
Messi's persistence then saw him squirm into the area and strike a low cross that was turned into his own net by Borja Gomez in time added on to destroy Granada's valiant resistance.
Barca moved on to 15 points from five games, opening an 11-point gap over champions Real Madrid who visit city rivals Rayo Vallecano on Sunday looking for their first away win this term.
Real Betis rose to third with nine points after a thumping Paulao header earned a 1-0 home win over misfiring Espanyol.
The Brazilian defender connected with a freekick after 18 minutes but his side failed to make the most of a second yellow card for Espanyol's Wakaso Mubarak in the 68th, and endured some shaky moments before securing the victory.
Earlier, South Korea striker Park Chu-young scored with his first touch on his home debut for promoted Celta Vigo, leading them to a 2-1 victory over Getafe.
Park, who helped his country win bronze at the Olympic Games in London and joined Celta on loan from Arsenal on transfer deadline day three weeks ago, volleyed home in the 68th minute.
Celta had taken the lead in the 57th with a breakaway goal from Augusto Fernandez, only for Abdelaziz Barrada to level almost immediately after a well-worked move.
Park, who netted two minutes after coming off the bench, had the chance to score a second near the end but he pulled his shot wide with only the keeper to beat.
Celta's second win put them ninth in the standings with six points from five games, two places ahead of Real Zaragoza, who won 3-1 at home to bottom side Osasuna.
Osasuna finished seventh last season but have only taken one point from 15 this term and trail Espanyol who also have one point. But they did score the goal of the game at the Romareda with Emiliano Armenteros's run and shot after 29 minutes.