Surya smashed an unbeaten 111 off only 51 balls. He hit 11 fours and seven sixes.
Southee said Mitchell would come back in, most likely for Mark Chapman, on Saturday.
Tim Southee took five wickets as New Zealand bowled out South Africa for 111 in their second innings, a paltry total that was actually better than their first innings haul.
Shreyas Iyer followed his first-innings hundred with a crucial 65, forging 50-plus stands with Ravichandran Ashwin and Wriddhiman Saha to revive India who declared their second innings on 234/7.
Openers Will Young and Tom Latham mounted a strong reply for New Zealand who finished day two on 129 for no loss, 216 behind India's first innings total of 345.
Openers K L Rahul (65) and captain Rohit Sharma (55) added 117 off just 80 balls to set up the Indian win.
Both sides have made a number of changes with India skipper Rohit Sharma strengthening the bowling unit with the addition of Bhuvneshwar Kumar and Mohammed Siraj.
Tim Southee will captain the T20I side.
During last week's IPL match between Delhi Capitals and Kolkata Knight Riders, which is captained by Morgan, Ashwin tried to take a run after a throw ricocheted off his batting partner Rishabh Pant's body.
In a series of tweets, Ashwin clarified his stand and made it clear that he would again run if the ball hit a fielder's body.