Set a Duckworth-Lewis target of 163 in 26 overs, the hosts survived anxious moments before winning with seven balls to spare at the Premadasa stadium to grab the lead in the three-match series.
Dilshan hit six boundaries and three sixes before Sri Lanka floundered while chasing the revised target after the West Indies, sent in to bat, had scored 159-8 in their 26 overs.
The hosts, cruising at 104-2, lost seven wickets for 48 runs before tailender Ajantha Mendis hit the winning six off Johnson Charles in the company of last man Suranga Lakmal.
Dilshan put on 46 for the first wicket with Kusal Perera and 43 for the second with Lahiru Thirimanne to give Sri Lanka a bright start.
But spinner Sunil Narine claimed three wickets, and Carlos Brathwaite and Jonathan Carter took two each, to cause a sensational Sri Lankan collapse when all seemed lost for the West Indies.
Earlier, Andre Russell and Darren Bravo put on 58 runs off 37 balls to steer the West Indies to 159-8 after they slipped to 24-3 before the rain break and then became 42-4 soon after resumption.
Skipper Jason Holder then hit 36 off 13 balls with three sixes and two boundaries as the West Indies plundered 59 runs in the final five overs.
Lakmal claimed the first three wickets and spinner Mendis took two.
Sri Lanka swept the preceding Test series 2-0. The second one-dayer will be played at the same venue on Wednesday.
The one-day series will be followed by two Twenty20 games.
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