Ireland, coached by former West Indies batsman Phil Simmons, collapsed to eight for two in reply, losing both openers cheaply, before Joyce, the captain of English county side Sussex, and Andrew Poynter steadied the innings.
Joyce's runs came off 49 balls, and it was left to Kevin O'Brien - just one of two survivors from the Ireland team that beat Pakistan at Sabina Park in 2007 - to hit the winning runs as the Irish finished on 117 for four with five balls to spare.
"It was obviously brilliant," said Joyce as Ireland took a 1-0 lead in the two-match series ahead of tomorrow's second T20 at Sabina Park and a standalone 50-over one-day game there on Sunday.
"It was a bit of a strange atmosphere but the nice thing is that we weren't jumping around like we'd won the World Cup.
"The bowlers set it up and we are all thrilled."
West Indies captain Darren Sammy said his men had been beaten by the "better team".
"Ireland played well and deserved their victory," the all-rounder added.
"We were a bit rusty and they came hard at us and in the end were the better team on the day."