The Dutchman suffered the injury during the first half at the Emirates Stadium and tried to play on before being withdrawn.
The problem is set to keep the 26-year-old out of Swansea's next five games, including the second leg of the Capital One Cup semi-final against Chelsea.
Manager Michael Laudrup said: "We had bad news after losing the other day, we have lost Kemy Agustien for, I think, the next four weeks with his ankle.
"He twisted it in the first half, he tried to start the second half and there was some problem with the ligaments.
"It could have been a lot worse, but still four weeks means he will be out for at least our next five games, four in the league and the game on Wednesday (against Chelsea) as well."