Costa crocked for Arsenal showdown
"The injury is expected to keep the player out of action for two weeks," Shakhtar said on their website.
The 20-year-old Brazil international injured his thigh muscle in Sunday's 2-0 league win over arch-rivals Dynamo Kiev.
Costa, who scored in Shakhtar's 3-0 win over Portugal's Braga in last week's Champions League match, will miss the top-of-the-table Group H clash against Arsenal in London on October 19.
Costa will also miss Brazil's friendlies against Iran and Ukraine over the next week.