Matt Henry and Dean Brownlie drop out of the squad from the Bangladesh series while Patel, who made a Test comeback after a gap of nearly four years against India last year, has been selected ahead of legspinner Ish Sodhi. Neil Wagner has also been confirmed as available following his finger injury he sustained in early February.
Though it is unlikely that New Zealand will field two spinners at University Oval - they have never done it in seven Tests at the venue - selector Gavin Larsen said Patel provided cover for all eventualities. He was brought into the one-day side in Hamilton, with a view to targeting South Africa's left handers, and claimed Quinton de Kock first ball.
"Jeetan brings the ability to spin the ball away from the left-handers, which complements our other spinner in Mitchell Santner nicely. He brings a huge amount of experience to the team and gives us the balance of having two front-line spin options."
Neesham, meanwhile, provides competition for the allrounder's spot alongside the incumbent Colin de Grandhomme who claimed six wickets on debut against Pakistan but has since taken just four wickets in six innings.
Neesham made an unbeaten 71 in the second ODI against South Africa in Christchurch and in his previous Test hit 71 against India at Indore before being dropped in favour of de Grandhomme at the start of New Zealand's home season.
"Jimmy [James Neesham] has impressed so far this summer and we've seen improvement in his game," Larsen said. "He gives us another option and is also very familiar with the conditions at University of Otago Oval."
Wagner's availability is a major boost for New Zealand. When he initially damaged his finger fielding off his own bowling for Otago early last month there were concerns he would not recover in time, but he played club cricket last weekend and came through unscathed. Since the beginning of 2016, he is New Zealand's leading Test wicket-taker with 51 scalps at 22.33.
"Having Neil back and available is fantastic news for us. Neil's been our leading wicket-taker in recent times and is a key player for us," Larsen said. "He's worked extremely hard to get himself ready for the series and we expect him to be fit and ready to go come the first Test."
BJ Watling was advised three weeks rest with a knee niggle last month but made 42 and 96 in last week's Plunket Shield match against Otago.