Koeman celebrated his 52nd birthday with three points, but saw Forster replaced after 10 minutes by veteran Kelvin Davis, who played an important role in Southampton's victory.
Davis made first-half saves from Sam Vokes and Danny Ings before Shane Long gave the home side the lead.
A Jason Shackell own goal wrapped up the win for Southampton, who remain in the hunt for a top-four finish, while Burnley missed a chance to climb out of the bottom three.
But it was the injury suffered by Forster in a minute collision with Vokes as he kicked the ball clear that gave Koeman and England manager Roy Hodgson cause for concern.
"Fraser Forster is at hospital and it is looking bad," Koeman told BBC Sport.
"We have to wait for the scan. I hope it will be OK for him but the doctor is not positive."
Forster looks set to miss upcoming games against Lithuania and Italy, but Koeman was thankful for the performance of experienced replacement Davis.
"It is good to have an experienced goalkeeper on the bench and he showed his qualities," added Koeman.
"It was a difficult first half, they had one or two very good chances but Kelvin was saving them.
"The second half was very good, we dominated it with good ball possession. A good win."
