Mathews scored 55 off 77 balls to help Sri Lanka put up a 200-plus total after Mahela Jayawardene's (52 off 52 balls) early enterprise at the Sabina Park.
Spinner Narine was the wrecker in chief for the hosts, returning impressive figures of 4/40 in 10 overs, as Sri Lnaka were bundled out in 48.3 overs.
Narine was well supported by Ravi Rampaul, who finished with 3/38, after West Indies skipper Dwayne Bravo opted to bowl on winning the toss on overcast conditions.
Pacer Rampaul then came back for his second spell to remove Lahiru Thirimanne and Nuwan Kulasekara as Sri Lanka slumped to 159 for six in the 38th over.
Earlier, Jayawardene hit seven boundaries during his run-a-ball knock, while Mathews found the fence five times.
After being sent into bat, Tharanga had made 25 off 43 balls before Bravo had him caught behind with a short of a length delivery that angled across the left-hander in the 14th over of the innings.
While his first boundary looked patchy, Jayawardeene got into the groove with a typically elegant four between mid-on and mid-wicket off an overpitched ball by pacer Kemar Roach.
The next one, off the same bower, was dispatched through the deep extra cover region.
