Stirling stood in a match-defining 106-run stand with Ed Joyce for the second wicket as Ireland chased down a 305-run target with 25 balls to spare.
Stirling cracked his 92 off 84 balls, Joyce scored 84 off 67 and Niall O'Brien chimed in with an unbeaten 79 off 60 deliveries as they cashed in on wayward West Indies bowling.
It was Ireland's first win over the West Indies in six official matches and the first victory at this World Cup achieved by a team batting second.
The 12th-ranked Irish paid scant regard to the high-profile West Indians and enhanced their own reputation as giant-killers after beating other top-tier nations England and Pakistan in the previous two World Cups.
The victorious finish was a fitting reward for their spirited start to the match when they opted to bowl and rapidly reduced the West Indies to 87-5.
Chris Gayle again failed to fire as spinner George Dockrell carved through the upper order before a dashing century by Lendl Simmons and a personal-best 89 from Darren Sammy put some backbone into the West Indies innings.
Their power batting lifted the West Indies to what should have been a respectable 304 but they lacked the discipline in their bowling to defend the total.
Captain Jason Holder opened with a first ball that raced wide of slips to the boundary to give Ireland an immediate five runs from extras which started Stirling and William Porterfield on their way to a 71-run first wicket stand. With Porterfield's dismissal for 23, the left-handed Joyce joined right-hander Stirling to continue piling on the pressure.
Stirling's dismissal, caught behind off a Marlon Samuels delivery, saw O'Brien arrive to up the tempo further with the textbook formula of looking for an early boundary each over and accumulating what runs were on offer afterwards.
The West Indies were powerless to contain the onslaught, going through eight bowlers, as the Irish batsmen clobbered six sixes and 34 fours.
Joyce, on 42, was dropped by Darren Bravo when he belted Holder towards the boundary, and O'Brien was on 38 when he skied a Jerome Taylor delivery and Holder dropped what should have been a regulation catch. Taylor was the most successful West Indies bowler with three for 71.
Ireland started the day in the best possible fashion, winning the toss and ripping out the West Indies premier batsmen.
Dockrell, who finished with three for 50, claimed the prized wickets of Gayle (36), Samuels (21) and Ramdin (one) in the space of just eight deliveries.
It was another disappointment for Gayle who has not passed 50 since playing Bangladesh in August and the last of his 21 centuries was against Sri Lanka in June 2013.
Simmons and Sammy gave the innings respectability with a West Indies sixth-wicket record stand of 154 Simmons — whose uncle Phil, the former West Indies batsman, is Ireland's coach — was eventually dismissed in the last over for 104, his second one-day international century
