After winning the toss and deciding to bat, Brandon King scored an unbeaten 62 as West Indies posted 146-5.
Andy Balbirnie's side, who had recovered from a 31-run defeat to Zimbabwe by beating Scotland, reached their victory target with 15 balls to spare as Paul Stirling made a fine half-century.
While Ireland go on to the main Super 12s stage, the two-time champions are left with an ignominious first-round exit.
Ireland's spinners helped restrict the West Indies attack, with Gareth Delany taking three for 16 and Sami Singh one for 11 from his two overs.
King, brought back into the side in place of Sharmarh Brooks, made his 62 not out from 48 deliveries, hitting six boundaries and one over the rope in what looked decent batting conditions at the Bellerive Oval.
Despite Odean Smith adding a rapid 19 late in the innings, the total always looked short.
Ireland soon set about their run chase, with openers Stirling and Balbirnie making 73 for the first wicket before the skipper was caught at point off Akeal Hosein for 37.
Lorcan Tucker continued to support Stirling, making 45 not out as he hit the winning runs to seal a memorable victory.