The England international was withdrawn shortly after half-time of Monday's 2-1 victory, and will now miss Premier League matches against West Ham, Newcastle United and Liverpool, and next week's UEFA Champions League last 16 second leg with Monaco.
Arsenal boss Arsene Wenger also stated that the injury will rule Oxlade-Chamberlain out of England's matches against Lithuania and Italy later this month.
"Alex has a classic hamstring which means he will be out for three or four weeks maximum and will certainly not be available for England," Wenger said.
"It's frustrating for him but he's played more games this season than before. He's shown his importance."
Wenger also stated that England's games will come too soon for midfielder Jack Wilshere who remains absent with an ankle injury.
He added: "It's very difficult to say when he will be back. We have to respect his recovery."
However, the Frenchman had better news on captain Mikel Arteta, who has not featured since November due to an ankle problem, saying the Spaniard could be back by the start of next month.
"He's back in full training," Wenger added. "I think he'll return in three weeks."
Wenger also confirmed that Mathieu Flamini is available for Saturday's clash with West Ham after a month-long layoff with a hamstring injury, while centre-back Gabriel Paulista will return to training on Sunday.
