Sri Lanka won the toss and chose to bat at the P Sara Stadium in Colombo, where numerous third umpire reviews before lunch left the hosts reeling after they had won the first Test by an innings.
Milinda Siriwardana, playing in only his second Test, stood firm against the West Indies attack, notching up his maiden Test 50 including two sixes.
But the visitors took two wickets before the break, including that of Dinesh Chandimal whose century in the first Test in Galle proved instrumental in Sri Lanka's win.
Warrican and captain Jason Holder were the pick of the bowlers with two wickets apiece.
In the first session before lunch, the visitors tasted early success when Taylor had opener Kaushal Silva caught behind off the fourth ball for a duck.
Umpire Rod Tucker was forced to reverse his initial decision of not out after Holder called for a review.
Opener Dimuth Karunaratne, who scored 184 in the first Test, looked shaky from the start in the face of the visitors' attack.
A third review saw series debutant Kusal Mendis, who made an unimpressive 13 in his first Test innings, given out for a faint edge off Kemar Roach.
After a fourth review, captain Angelo Mathews was also given out, caught by Kraigg Brathwaite at second slip off Holder for 14.
The West Indies have not found Sri Lanka a happy hunting ground since they played their first official Test on the island in 1993.
Now they are next to bottom -- above only Bangladesh -- in the league table of regular Test-playing nations while Sri Lanka are one notch above them in seventh place.