Patel made it two wickets in two overs as Shubman Gill, who had scored a splendid 90 in the first innings, left a ball that crashed into the stumps and departed for one.
Virat Kohli did not last long after arriving at the crease as he edged Patel back to slip on one to leave India in trouble at 18-3 on a tricky Wankhede Stadium pitch.
The crowd fell silent again as Yashasvi Jaiswal was trapped lbw for five by Glenn Phillips and Sarfaraz Khan hit a full toss from Patel straight to Rachin Ravindra in the deep on one.
Ravindra Jadeja steadied the ship alongside Pant in a 42-run stand, but a stunning catch by Will Young meant India were on the ropes at 71-6.
Patel dismissed Pant following a New Zealand review after lunch, although the batsman appeared to suggest that he had not made contact with the ball before it was caught.
Loading
The runs dried up after that and India collapsed in a heap, with Washington Sundar the last to go while looking to go big.
“Losing a series, losing a Test match is never easy. It’s not easily digested,” said Sharma. “But we didn’t play our best cricket and we accept that. New Zealand played better than us throughout.
“There were a lot of mistakes that we made, we accept it ... As a captain, I wasn’t at my best leading the team and with the bat as well.”
New Zealand resumed their second innings on 171-9 earlier, but India needed only 14 balls to bowl the Black Caps out, as Jadeja had Patel caught in the deep to end with fine figures of 5-55 after also bagging five in the first stanza.
The tourists won the opening match in Bengaluru by eight wickets for their first Test victory in India in 36 years, and wrapped up the series in Pune with a 113-run win.
New Zealand’s maiden series triumph in India going back to 1955 also snapped the hosts’ home streak of 18 straight series triumphs since a 2-1 loss to England in 2012.
Reuters