They surpassed the previous record of 263-5, set by Royal Challengers Bangalore in 2013
The 29-year-old Dhawan hit 134 and 28 with the bat but did not take the field during Sri Lanka's second innings in the match that the hosts won by 63 runs to take a 1-0 lead.
"Dhawan had injured his right hand during the first Test at Galle," the BCCI said in a statement.
"He underwent further Tests that confirmed a hairline fracture in his hand. He would require four to six weeks to recover from the injury."
The board did not name a replacement for Dhawan but added all-rounder Stuart Binny to the squad on Sunday.