Dilshan hit eight fours on his way to notching 56 after the West Indies won the toss and opted to field at the Pallekele International Cricket Stadium in Sri Lanka.
The start of the match was delayed by over an hour due to rain, and when play finally got under way all of Sri Lanka's top-order batsmen were in fine form.
Left-handed opener Kusal Perera scored 40 before being caught by Darren Sammy off burly bowler Kieron Pollard for the fall of the first wicket.
Shehan Jayasuriya was on 36 when he was caught and bowled by Sunil Narine.
At the end of the 20 overs Dinesh Chandimal was on 40 while Angelo Mathews was 37 not out, leaving the West Indies chasing a difficult target of 216.
The two-match T20 series comes after Sri Lanka swept the three-match one-day international series 3-0.
Sri Lanka won the preceding Test series without loss, taking it 2-0.
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