The home side eventually chased down the 132-run target with 41 balls to spare, taking a 1-0 lead with a four-wicket win.
Dananjaya became only the 15th player and the fourth Sri Lankan to take a T20 international hat-trick when he dismissed Evin Lewis (28), Chris Gayle (0) and Nicholas Pooran (0) off consecutive deliveries in the fourth over.
The leggie had little time to reflect on his fame as he was hit to all parts of the ground in his next over by Pollard, who became only the second player after India’s Yuvraj Singh to strike six sixes in an over in a T20I.
Dananjaya experienced see-sawing fortunes as he was hit for a seventh consecutive maximum by Jason Holder (29 not out off 24 balls) in his next over and had two catches dropped off his bowling.
Batting first, Sri Lanka managed only 131 for nine. The away side was 86 for three at one stage, but fell apart after a short rain break and lost six wickets for just 45 runs.
BRIEF SCORES: Sri Lanka 131 for 9 in 20 overs (P Nissanka 39) lost to West Indies 134 for 6 in 13.1 overs (K Pollard 38, W Hasaranga 3/12, A Dananjaya 3/62)