The match had been reduced to 44 overs per side after a rain-delayed start at the Pallekele International Stadium.
Asalanka, the captain, had won the toss and his team bowled West Indies out for 189 in 36 overs. In reply, Sri Lanka scored 5-190 in 38.2 overs.
Sri Lanka had earlier reduced the tourists to 8-58 but a valiant 119-run partnership between Sherfane Rutherford and Gudakesh Motie for the ninth wicket brought them to a respectable score.
Rutherford scored 80 off 82 deliveries and hit four sixes and seven boundaries. He benefitted from two dropped catches, by Janith Liyanage and Kamindu Mendis.
Motie, batting at No10, made his maiden international half-century and remained unbeaten on 50 off 61 deliveries including six boundaries.
Leg spin bowler Wanindu Hasaranga had the best bowling figures for Sri Lanka with 4-40. Spinner Maheesh Theekshana and seamer Asitha Fernando picked three wickets each.
West Indies bowlers took two early wickets during Sri Lanka's run chase having them two down for 25. But opener Nishan Madushka (38) and Sadeera Samarawickrama (38) shared 62 runs for the third wicket to stabilise the innings.
Another 55-run partnership between Asalanka and Janith Liyanage (24) carried Sri Lanka closer to victory. Asalanka made 62 not out off 61 deliveries including a six and seven boundaries.
Seamer Alzarri Joseph had 2-30 for the West Indies.
The final match will be played on Saturday at the same venue.
