Put into bat by England, West Indies chalked up 289-6 in its 50 overs with middle-order batsman Hetmyer smashing an unbeaten 104 from 83 balls as Windies scored 81 in the last ten overs.
"My celebration was for my girlfriend," Hetmyer said." I try to do good things whenever she is around. The guys will be hoping she's always around!"
Veteran opener Chris Gayle also chipped in with 50 from 63 deliveries, while Shai Hope made 33.
England made a terrible start to its reply, slumping to 60-3, but captain Eoin Morgan and Ben Stokes staged a fightback with a fourth-wicket partnership of 99.
Morgan hit 70 and Stokes made 79 before both fell in quick succession, and the rest of the batting failed to deliver as the tourists were bowled out for 263 in 47.4 overs.
Cottrell took 5-46 and fellow paceman Jason Holder returned figures of 3-53.
"We made mistakes, it hurt us and the West Indies deserved to win," Morgan said.