Chasing a target of 348, West Indies were bowled out for 160 before tea on the final day.
Sri Lanka secured a 156-run first-innings lead and set the visitors a mountainous 348 to win at the Galle International Stadium.
Reeling on 52 for six at the end of day four, the West Indies clung on in the opening session but collapsed after lunch.
Lasith Embuldeniya finished with a five-wicket haul while Ramesh Mendis complemented him well with four scalps.
Nkrumah Bonner was unbeaten on 68 for the tourists — one of only three West Indies batsmen to make it into double figures in their second innings, the others being Joshua Da Silva, who made 54, and Rahkeem Cornwall with 13.
The second Test begins on Monday, also in Galle.
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