Oram's replacement in the squad of 12 is Chris Cairns, who is being pushed into the side earlier than planned after recovering from two knee operations this year.
Chairman of selectors Richard Hadlee said if Cairns was chosen to play in the 11, he would not be required to bowl. It had been envisaged that Cairns's comeback match would be the sixth one-day international in Auckland on Saturday.
New Zealand manager Jeff Crowe said Oram was injured when he attempted to take a low catch and was struck on the end of his left little finger. A hospital x-ray revealed the break.
"Jacob is to be examined by an orthopaedic surgeon this evening, who will assist the New Zealand Cricket medical team in determining the best recovery programme for him," Crowe said.
Crowe said he was confident Oram's injury would not prevent him flying to South Africa with the World Cup squad on January 29.
Oram has been one of New Zealand's best performers in the seven-match series, which the home side lead 4-0.
He was man of the match in the opening game in Auckland with bowling figures of five for 26 and finishing 27 not out as he nursed New Zealand home from a precarious 52 for six to 109 for seven and victory by three wickets.
Oram is a right-arm medium fast bowler and left-handed batsman. He returned to the national side after a season out with injury. He now has a one-day career of 26 wickets at 31 and 377 runs at 16.3 from his 29 matches.
