The full-back limped off during the game and had to be replaced by Andre Santos.
"Gibbs has a thigh strain and will be out for three weeks. It's a big blow for us," Gunners boss Arsene Wenger told reporters after the game.
"He was an important player for us going forward and the game we wanted to play. It's a shame he's out for the next three weeks."
The news has sparked rumours that Wenger will move to strengthen his defence before the transfer window closes tomorrow.