The England striker scored his seventh league goal of the season from the penalty spot, despite receiving earlier treatment for the knock.
Cole, who was signed from Chelsea in July 2006, had just returned from a hamstring strain against Aston Villa on 4 November.
After him, West Ham's highest league scorer this season is Mexican Guillermo Franco, who has netted three times, so his firepower will be missed over the festive period.
West Ham, who lie in 17th position, have some tough matches in the remainder of 2009 as they attempt to move clear of the relegation zone.
The home match against Manchester United is followed by trips to Birmingham and Bolton, home games against Chelsea and Portsmouth, and an away fixture at Tottenham.
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