Nehra won the approval after his impressive effort in the second Hamilton Test, although India lost the match by four wickets earlier on Sunday to concede the two-test series 2-0.
The selectors had picked 14 players on Wednesday but decided to watch the performance in Hamilton before filling the final two slots.
The 24-year-old Patel has been rewarded for his consistency at domestic level in the last two seasons and has also been included among the 30 probables for next year's World Cup.
Fast bowler Javagal Srinath, leg-spinner Anil Kumble and batsman Yuvraj Singh will also join the rest of the squad in New Zealand.
Batsman Shiv Sundar Das, fast bowler Tinu Yohannan, left-arm spinner Murali Kartik and wicket-keeper Ajay Ratra will return home, a Board of Control for Cricket in India release said.