Suryakumar, captaining India for the first time, hit an assured knock of 80 as the hosts chased down 208 in Visakhapatnam and secured a small measure of revenge for their painful defeat by Australia in the 50-over World Cup final on Sunday.
Ishan Kishan (58) and Suryakumar shared a 112-run partnership, but India lost momentum after both players departed and the chase went down to the final ball of the match when Rinku Singh guided his team over the line.
"Credit goes to (Suryakumar) for the way he played. He was taking on the bowlers," Kishan said.
"The communication was very good. I'm really happy because it's a young side. We'll make mistakes, but hopefully we'll come back stronger as a bowling unit."
Australia had racked up an impressive total after being put in to bat after Inglis hit 11 fours and eight sixes in his 110 from 50 balls.
His century came from 47 balls, tying Aaron Finch's Australian record.
Inglis was ably supported by Steve Smith (52), another of the seven players from Australia's World Cup-winning squad to remain in India for the T20 series, in a 130-run second-wicket partnership.
The second match in the five-game series is in Thiruvananthapuram on Sunday.
