After winning toss and electing to bat first, India were in a spot of bother at 111/7 but Bumrah's maiden first-class fifty took India to a respectable 194.
Bumrah posted the highest score of the innings with an unbeaten 55 from 57 balls, including six fours and two sixes and shared a 71-run last-wicket stand with Mohammed Siraj (22) to lift India from 123-9 in the 36th over to 194 all out in the 49th.
Shubman Gill (43) and opener Prithiv Shaw (40) were the two other main contributors for India.
Sean Abbott 3 for 46 and Jack Wildermuth returned 3 for 13 for Australia A, while Will Sutherland, Mitch Swepson, Harry Conway and Green took one wicket each.
In reply, India came back strongly with the ball to bundle out Australia A for 108 as Shami (3/29), Bumrah (2/33), and Saini (3/19) bowled a fiery spell. India lead by 86 runs.
Captain Alex Carey (32 off 38) was the top scorer for the home side.
