In a statement, NZC said that the decision on Jamieson was made after he “experienced stiffness in his hamstring following his arrival home from Bangladesh last week”.
Allen will miss just the one ODI and will rejoin the squad for the third game.
The second ODI will be played in Nelson on Wednesday.
New Zealand squad for rest of ODI series: Tom Latham (capt), Will Young, Tom Blundell (wk), Henry Nicholls, Mark Chapman, Josh Clarkson, Rachin Ravindra, Adithya Ashok, Jacob Duffy, Adam Milne, William O’Rourke, Ben Sears, Finn Allen (only third ODI)