Though they poached an early scalp, the Sri Lankans were put on the back-foot as West Indies reached 96 for one in their second innings when stumps were drawn on the fourth day at the Guyana National Stadium.
Malinda Warnapura scored 62 and Thilan Samaraweera hit an unbeaten 56 that propelled Sri Lanka to 240 for seven before they declared for the second time in the match, after West Indies managed to avoid the follow-on, but conceded a first innings lead of 196 when they were dismissed for 280.
The Sri Lankans were then put on the defensive by enterprising batting from makeshift opener Dwayne Bravo, in particular, and West Indies vice captain Ramnaresh Sarwan.
The visitors failed to find an answer for the positive stroke-play of the pair, as they added 74 - unbroken - for the second wicket.
The Sri Lankans were jumped when Bravo emerged with left-hander Devon Smith to open the batting instead of West Indies captain Chris Gayle, obviously being shielded from the early misery of Chaminda Vaas' expert left-arm swing bowling.
The visitors saw the pair play some audacious strokes before Vaas had Smith caught at deep backward square leg for 10 by a sprawling Thilan Thushara when the batsman miscued a hook at a short, rising delivery.
Sri Lanka's bowlers produced some nervous moments for Bravo and Sarwan, but the two West Indies batsmen consolidated to carry the home team through until the close.
Before lunch, Sri Lanka failed to make West Indies follow-on, after the hosts continued from their bedtime total of 269 for nine.
The visitors knew they had to bat a second time, when Daren Powell edged Muttiah Muralitharan past Sri Lanka captain Mahela Jayawardene at slip to third man for three.
After Muralitharan and Vaas, Sri Lanka's two most experienced bowlers, failed to separate the two West Indies tail-enders, Jayawardene turned to Thilan Thushara and he responded with the scalp of Powell caught low down at second slip for 12, leaving Taylor not out on 27.
Thushara, playing his first Test since making his debut on Sri Lanka's previous trip to the Caribbean five years ago, ended with three wickets for 59 runs from 20.5 overs, Vaas took three for 48 from 25 overs, and Muralitharan snared three for 112 from 40 overs.
When Sri Lanka batted a second time, they gained another stable foundation from Warnapura and fellow left-handed opener Michael Vandort.
Sri Lanka however, suffered a setback in the final half-hour before lunch, when Vandort was caught down the leg-side for 24 off West Indies captain Chris Gayle, bowling his uncomplicated off-spin.
After the interval, the hunt for quick runs continued, but the Sri Lankans lost wickets at regular intervals, and were 186 for five at tea.
Warnapura, one of two century-makers in the Sri Lanka first innings, continued in his rich vein of form, and reached his 50, when he edged Powell for the last of his five boundaries through the vacant slip region in the first half-hour after lunch.
But the left-handed opener was one of four Sri Lanka wickets which fell between lunch and tea.
Sri Lanka had suffered a setback early in the afternoon period, when Kumar Sangakkara was caught at mid-wicket off Bravo for 21, following a second wicket stand of 51 with Warnapura.
Mahela Jayawardene came to the crease to pair the two first innings century-makers, but Warnapura, growing increasingly adventurous, was caught behind off Bravo chasing his ill-directed 91st delivery to end his innings which lasted a little over two hours.
With the Sri Lankans looking to increase the tempo, their batsmen played a few ill-advised strokes, and Jayawardene perished, when he was caught at mid-wicket for 33 top-edging a pull to become left-arm spin bowler Sulieman Benn's first Test wicket.
In the final half-hour before the break, Tillakeratne Dilshan was adjudged lbw to Taylor for four to leave Sri Lanka on 171 for five, and Samaraweera and Vaas to carry the visitors to tea.
After tea, Samaraweera led the charge for quick runs, and reached his 50, when he drove Benn to long-on for a single.
But Vaas was caught behind off Benn and Thushara was caught at long-off off the same bowler before the Sri Lankans felt they had enough.
Benn, playing his first Test, ended with three for 59 from 13 overs, and Bravo took two for 54 from 14 overs.
Sri Lanka are aiming to win their first Test match in the Caribbean having lost two and drawn two of their previous four here. The visitors are also eyeing a Test series victory for the first time in the Caribbean.