Anya Shrubsole accounted for four victims to help England bowl their opponents out for a meagre 101 - which remarkably included 23 wides - before Danielle Wyatt (40) led England to 103 for four.
Wyatt and Arran Brindle, who took three wickets for no runs in the West Indies innings, put on 66 for the first wicket and England survived losing three batters in quick succession to book their place in the next stage.
Brindle edged Deandra Dottin to keeper Merissa Aguilleira to depart for 14 and end the first-wicket stand. Sarah Taylor was trapped lbw by the next ball and Lydia Greenway, having survived Dottin's hat-trick ball, perished in similar fashion for two.
Wyatt's 80-ball innings ended when she was trapped in front by Subrina Munroe, having hit six boundaries, but Heather Knight (18 not out) and Jenny Gunn (4no) saw England home.
Shrubsole earlier took four for 21 to restrict the Windies, who were grateful for a partnership of 58 for the seventh wicket between Kyshona Knight (33) and Shanel Daley (30 no).
Despite winning the toss, the Windies struggled to prosper on a seam-friendly surface and they slipped to 31 for six, with Shrubsole and Katherine Brunt making the early breakthroughs.
Charlotte Edwards' side were then held up by Knight and Daley's fifty partnership although neither looked to take the attack to England, with both women going at less than a run every two balls.
Brindle made the breakthrough when Knight was stumped, and the 31-year-old then bowled Shaquana Quintyne for a double-wicket maiden.
Brindle had her third wicket in the next over before Shrubsole claimed her fourth scalp in the 37th over, with West Indies losing their last four wickets for just 12 runs.