Both players received treatment by medical staff almost immediately but Conway was "taken to A&E for further examination which confirmed a concussion", a statement by the New Zealand cricket board said.
Conway now joins Mark Chapman and Rachin Ravindra, who were both ruled out of the ODI series, on the casualty list with Tim Seifert set to be included in the XI.
New Zealand captain Mitchell Santner said: "Yeah, I guess he's (Seifert) the back-up batter in our squad. So it'll probably be a direct swap if Dev's unavailable, which is a shame.
"He (Conway) is obviously very good in one-day cricket and, you know, was biding his time in the T20 series. And, yeah, I guess another one of those freak accidents that happens."
India won the preceding five-match T20I series 3-2, although the Black Caps did win the final two games in the shortest format and will take momentum into the three ODI contests.
New Zealand's bowling unit fought hard to defend a low total but India came out on top by four runs (DLS method) in the second ODI in Wellington as the visitors went 2-0 ahead in the five-match series.
New Zealand fast bowler Jacob Duffy believes Virat Kohli remains one of the most difficult batters to contain, after witnessing the Indian star produce a match-winning performance at Eden Park.
Arshdeep Singh took four wickets before Virat Kohli hit 74 as India beat New Zealand by five wickets in a rain-affected first of five ODIs that was played in Auckland on Wednesday.
Glenn Phillips struck a century while seamer Jacob Duffy took four wickets as New Zealand claimed a consolation victory in the T20I series against India in Auckland.
