The opening match was washed out last week.
The 28-year-old Dawson, who picked up the wicket of wicketkeeper-batsman Kusal Perera in the match at Dambulla, is scheduled to return home on Tuesday.
The ECB said Kent all-rounder Joe Denly, who was named in the test squad for the tour, has been drafted into the ODI squad as Dawson's replacement.
Denly, who bats at the top of the order and bowls leg-spin, is expected to join the squad in time for the third ODI at Kandy on Wednesday.
England already have two other spin options in Moeen Ali and Adil Rashid.
Following the ODI series, England take on the hosts in a one-off Twenty20 international match on Oct. 27. The teams then play three tests in November.
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Pritha Sarkar)
