Allen has been ruled out of the tournament after suffering a foot injury while representing the San Francisco Unicorns in the Major League Cricket competition in the United States.
"We're really gutted for Finn," head coach Rob Walter said in a statement issued by the New Zealand association on Sunday.
"I was looking forward to working with him and to see him continue his form from the MLC, but unfortunately injuries happen.
"We're lucky to be able to call on someone of Devon's quality to replace Finn."
Mitch Hay, Jimmy Neesham and Tim Robinson will also join the squad as cover for Michael Bracewell, Mark Chapman, Glenn Phillips and Rachin Ravindra, who are involved in the MLC final on Monday.
"We knew there'd be a possibility that a handful of players could be involved in the MLC final on Monday, so we're bringing in Mitch, Jimmy and Tim as possible replacements," Walter said.
Hosts Zimbabwe face South Africa in the tournament opener on Monday night (Australian time).Â
New Zealand begin their campaign against South Africa on Wednesday.
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