Shai Hope and Justin Greaves formed an unbroken 174-run partnership for the fifth wicket.
West Indies reached 318 for four by the end of the third day, trailing Sri Lanka's 549/9.
Sri Lankan bowlers, including Prabath Jayasuriya, struggled to break the defensive stand in the afternoon.
Kavem Hodge and John Campbell provided early resistance, but Sri Lanka remained in a strong position.