Having recovered from a 31-run defeat to Zimbabwe by beating Scotland, Balbirnie's side used that momentum to pull off another impressive display.
Although Brandon King scored an unbeaten 62, Ireland's spinners helped restrict the batting attack, with Gareth Delany taking three for 16 and Sami Singh one for 11 from his two overs, as West Indies posted 146-5.
Openers Paul Stirling and Balbirnie then made 73 for the first wicket before the skipper was caught at point off Akeal Hosein for 37.
Lorcan Tucker (45 not out) continued to offer support for Stirling, who finished with a fine unbeaten 66 from 48 balls, including half-a-dozen boundaries and two sixes.
Ireland advanced to the next stage of the tournament for the second time, last making it through in 2009.
"This means everything to us. We were so disappointed to miss out at this stage last year, so we had to do a lot of thinking and make a lot of changes back home," Balbirnie said after Ireland's victory, reported by BBC Sport.
"We set out some things we wanted to achieve as a team and how we wanted to play.
"And here, we lost our first game, but to then come back and beat the two-time champions in a must-win game, I could not be prouder.
"It was an emotional finish, we are absolutely delighted."
Balbirnie added: "A lot of things can happen in T20 cricket when it's your day. We want to test ourselves against the best in the world and who knows where that can take us?"
While Ireland celebrate another milestone, West Indies, the two-time champions, are left with an ignominious first-round exit.
West Indies captain Nicholas Pooran, who was out for 13, said: "It's really tough.
"I want to say congratulations to Ireland, they batted and bowled fantastically today.
"We haven't batted well enough throughout the tournament. This is a learning experience for us, we are really disappointed and we know our fans will be disappointed back home.
"It's definitely hurting. I'm disappointed in myself and how it has gone but we will live to fight another day."