The 23-year-old striker, who is on loan from Premier League rivals Liverpool, had a great start to life at Upton Park, worrying the Cottagers' back-line throughout as the Hammers opened up a three-goal lead before half-time.
Carroll limped off in the second half with the injury and West Ham co-owner David Gold has now confirmed how long the striker is set to miss on his Twitter page.
Gold tweeted: "Six weeks out. Back mid October. I think this would be the worst case scenario. DG."
Hammers boss Sam Allardyce is under no illusion as to how important Carroll is to his side's chances this season.
He told reporters: "If Andy scores between 10 and 15 goals, it is going to give us a great chance to be out of the relegation zone.
"If Andy gets 15 to 20 goals, we can get in the top ten because all we have to do there is to produce enough clean sheets to make sure we get points on the board with the goals that he scores."