The left-arm spinner required four stitches on his left index finger after suffering the injury while attempting a difficult return catch from hard-hitting batsman Jos Buttler in the penultimate over of England's innings.
He left the field immediately with blood dripping from the wound.
"The doctor says he's out for 8-10 days, but we'll see how he heals up and then reassess," Sri Lanka team manager Michael de Zoysa was quoted as saying in a report on ESPN Cricinfo.
Sri Lanka beat England by nine wickets for their third victory in four matches and face co-hosts Australia in Sydney next Sunday.
Image: Rangana Herath
Photograph: Phil Walter/Getty Images