The West Indies closed the fourth day of the first Test on 96-1, having been set a victory target of 437 by Sri Lanka in Georgetown, Guyana, on Tuesday.
Sri Lanka bowled out the hosts for 280 in the morning and then scored 240-7 before declaring midway through the evening session.
Stand-in opener Dwayne Bravo, promoted ahead of the out-of-form Chris Gayle, closed on 46 not out, with Ramnaresh Sarwan unbeaten on 34.
West Indies, after a solid start, now need 341 runs to win the Test.
Chaminda Vaas struck an early blow as Devon Smith (10) top-edged an attempted pull stroke and was brilliantly caught at long leg by a diving Thilan Thushara.
In the morning, the West Indies, who started the day on 269-9, managed to avoid the follow on before Thushara had last-man Darren Powell caught at slip.
Thushara finished with 3-59. Vaas claimed 3-48 and Muttiah Muralitharan finished with 3-112.
Opener Malinda Warnapura, who scored a maiden Test century in the first innings, then top scored in the second innings with 62 from 91 deliveries with five boundaries.
Middle-order batsman Thilan Samaraweera also scored an unbeaten 56, while skipper Mahela Jayawardene made a brisk 33 from 38 runs as Sri Lanka searched for quick runs.
Warnapura and Michael Vandort started the innings with a brisk 43-run opening stand.
West Indies claimed one wicket before lunch as Vandort was well caught down the leg-side for 24 by wicket-keeper Denesh Ramdin off part-time spinner Gayle.
Kumar Sangakkara (21), trying to push along the scoring rate, departed shortly after lunch as he tried to clip a slower delivery from Dwayne Bravo through the leg-side and was caught at mid-off from a leading edge.
The breakthrough ended a 51-run stand for the second wicket.
Warnapura perished as he chased a wide delivery from Bravo.
Jayawardene kept the scoreboard ticking along before being surprised by some extra bounce from left-arm spinner Sulieman Benn and was caught at mid-wicket — Benn’s first wicket in Test cricket.
The West Indies grabbed their fourth wicket of the middle session when Jerome Taylor trapped Tillakaratne Dilshan lbw.
Benn finished with 3-59 from 13 overs, the most successful of the West Indian bowlers. — Reuters