Shami, India's pace spearhead, returned figures of 3-18 to help dismiss the Black Caps for 108 after the hosts elected to bowl first in Raipur.
India looked to be coasting to victory with New Zealand in deep trouble at 131-6 while chasing 350 after opener Gill made just the 10th double hundred in ODI history.
Left-hander Conway made 52 off 35 balls and Mitchell smashed an unbeaten 59 as the Kiwis posted 176-6 after being invited to bat first in Ranchi.
India reached 257 for six at stumps in response to England's 578 in Chennai. Washington Sundar, on 33, and Ravichandran Ashwin, on eight, were batting with an ubroken stand of 32. India need another 122 to avoid the follow-on.
Even as fans in almost all cricketing nations will be glued to TV during the India-New Zealand World Test Championship final (WTC) starting on June 18, Australian cricket enthusiasts will miss out as no broadcaster has secured rights to televise the match that will decide the world's best Test team.
New Zealand enjoyed the greatest triumph in their cricket history as they beat India by eight wickets in the inaugural World Test Championship final at Southampton on Wednesday.
When India take field in World Test Championship (WTC) final from June 18, they will be up against a poor record versus New Zealand in major International Cricket Council (ICC) events stretching back to 18 years. India’s seven-wicket win in the 2003 World Cup under Sourav Ganguly
The inaugural World Test Championship final suffered a setback at the outset after persistent rain meant there was no play at all between India and New Zealand on Friday’s opening day at Southampton. A match that represents the culmination
After the first day’s play was washed out due to rain, the inclement weather encouraged New Zealand skipper Kane Williamson to field first after winning the toss with five fast bowlers chosen in the playing XI.