The France international picked up the problem during the 2-0 win over Napoli in the Champions League on Tuesday, although he did manage to play the whole game.
The injury ruled Sagna out for the Sky Live clash with West Brom at The Hawthorns on Sunday.
He also looks set to miss league games against Norwich and Crystal Palace, as well as the Champions League match with Borussia Dortmund on 22 October.
"He will not be fit. He has a hamstring injury and will be out for the weekend and for basically three weeks," said Wenger, who is set to give England under-21 international call-up Carl Jenkinson another chance to impress.
"It is always a blow because he is a very complete defender. He can defend well, he is good in the air, he can attack well as well, and he's a winner-type, so of course for us it is a handicap
"But Jenkinson had a good start to the season as well and he is growing as a player. I am sure he will do very well."
Spanish playmaker Santi Cazorla is also out of contention for the West Brom game due to an ankle problem.
Wenger is expecting a stern test of his side's form this weekend, when Arsenal will be out to record an 11th straight win in all competitions against a side on the back of an impressive shock away win at Manchester United.
"They had quite a strong side out (in the Capital One Cup) and many of the regular players played, but they have a big squad," Wenger said.
"I noticed when I looked at their team who played at Manchester United, that they have a massive squad of experienced players, so it's very difficult to predict who will play against us."
Watch West Brom v Arsenal on Sunday from 3:30pm on Sky Sports 1 HD