Woakes took 5-19 in 7.3 overs of skilled swing bowling to dismiss the West Indies for a paltry 122 in 29.3 overs. England chased down the target for the loss of just one wicket in 22.5 overs.
Woakes dismissed dashing opener Chris Gayle off the third ball of the first over for a duck and off the very next ball had Darren Bravo, who also failed to score.
He then dismissed Dwayne Smith (21) in his fourth over to leave the West Indies reeling on 35-3.
Fellow fast bowler Steven Finn took 2-34.
Opener Moeen Ali hit 46 off 43 balls with nine boundaries and put on 70 for the opening wicket with Ian Bell who made 35 not out.
James Taylor made 25 not out and helped add 55 in an unbroken second-wicket stand.
The victory will give England much-needed confidence before their opening World Cup game against Australia in Melbourne on Saturday.
