Capoue had to be carried off on a stretcher after 72 minutes of the north London derby following a tackle from Gunners midfielder Santi Cazorla.
Manager Andre Villas-Boas feared the summer signing from Toulouse had broken his fibula, but Spurs announced on Monday afternoon that the Frenchman had only suffered a sprained ankle.
A statement on Tottenham's website read: "Etienne Capoue has this morning (Monday) undergone an MR scan which has revealed the midfielder suffered an ankle sprain against Arsenal on Sunday.
"The recent signing from Toulouse was injured after 72 minutes at the Emirates Stadium and was immediately taken to hospital to consult with a foot and ankle specialist.
"Early indications suggest the France international will be unavailable for a period of four weeks."
Villas-Boas lost defensive midfielder Sandro the second half of last season with a serious knee injury and Villas-Boas had feared he would be without 25-year-old Capoue for a similar length of time.
Should Capoue's rehabilitation go according to plan, the £8.5million signing will miss just three matches, with his return expected to come against Chelsea on September 28.
Given the injury is only a sprain, Villas-Boas is unlikely to dip in to the market for a replacement, especially as he has already brought in seven new players this summer.
Sandro started Tottenham's last two games on the bench, but he is likely to be drafted in to the starting XI for the next game against Norwich on September 14.
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- Etienne Capoue