Woakes took 5-19, including the wickets of Chris Gayle and Darren Bravo in successive balls in the first over, as West Indies were bowled out for 122 in 29.3 overs after electing to bat at the Sydney Cricket Ground.
Moeen (46 from 43 balls) then struck nine fours in a first-wicket partnership of 70 with Ian Bell in 11.5 overs, before Bell (35) and James Taylor (25) guided England home with 27.1 overs to spare.
Eoin Morgan???s side were without rested seamers James Anderson and Stuart Broad but the experienced pair were not missed, with Woakes making immediate inroads into the West Indies batting line-up.
The Warwickshire man took just three balls to strike, Gayle removed for a first-ball duck after gloving a leg-side catch to wicket keeper Jos Buttler.
Bravo then fell to the next delivery, getting a thin edge to a ball which was angled across him and providing Buttler with another catch.
Woakes produced another beauty with his hat-trick ball that almost caught the edge of Marlon Samuels??? bat, but the 25-year-old did snaffle a third wicket when he had Dwayne Smith (21) caught by Bell at second slip with the score on 35.
Samuels was clean bowled by Steven Finn (2-34) before Windies could add to their tally and the Caribbean outfit were 51-5 when Chris Jordan castled Denesh Ramdin (6).
Lendl Simmons (45 from 55) was the only batsman to impress, with a huge six of James Tredwell into the Members??? Pavilion ensuring West Indies reached three figures.
The Trinidadian received little support, however, and was eventually pouched at short cover by James Tredwell as Woakes claimed his fourth scalp of the day.
And Woakes ended Windies’ innings when he had Andre Russell caught in the deep by Jordan after Tredwell (1-20), Ravi Bopara (1-33) and Finn had pinched lower-order wickets.
Moeen then showed his intent in England???s reply by driving Kemar Roach through cover in the third over and continued to target the off-side during his 55-minute stint at the crease.
The left-hander should have departed in the fourth over when he cracked straight to point but with Jason Holder overstepping the Worcestershire star was handed a reprieve and he duly made West Indies pay.
Moeen finally departed clubbing Roach to Bravo at extra cover from the penultimate ball of the 12th over but Bell and Taylor – aided by West Indies giving up 19 extras – sealed England’s triumph in the 23rd over.
Morgan’s side will face Pakistan on Wednesday in their final World Cup warm-up, before meeting Australia in Melbourne on Saturday in their first match of the tournament.