Bracewell tore his calf in the one-day international series against India last month and recovered in time for the World Cup but damaged the muscle again during warm-ups ahead of Sunday's win over Afghanistan.
"We're all feeling for Michael. It's really tough to get ruled out of a World Cup and I know how much playing for New Zealand means to him," head coach Rob Walter said.
"Cole is a seasoned campaigner. He brings a great all-round skillset to the group along with a lot of T20 experience. It's great to be able to lean on an experienced head especially for a World Cup."
New Zealand beat UAE by 10 wickets on Tuesday to top Group D and next play South Africa on Saturday in Ahmedabad.