Electing to bat in the four-day game, the Aussies were hurt by pacer Umesh Yadav's (3/42) early burst that fetched him two wickets.
Yadav drew first blood after cleaning up opener Alex Doolan for 18 and soon got left-hander Phillip Hughes for 12. Mishra kept up the pressure to pack Chris Lynn (12) back to the pavilion.
Recovering from 64 for three, Peter Forrest (77) and skipper James Faulkner (94) then put on a 133-run fourth-wicket partnership to resurrect the innings.
Mishra (4/114) then spun into action and the veteran leg-spinner got Faulkner caught behind as the number-five batsman got his 11th first-class fifty but missed out on his maiden ton.
The unstoppable Mishra then got Mitchell Marsh for 14 and soon cleaned up wicketkeeper-batsman Sam Whiteman for 12 to break the back of the Aussie line-up.
Medium-pacer Jasprit Bumrah, who opened the bowling alongwith Yadav, pacer Anureet Singh and off-spinner Baba Aparajith also bowled well but were unlucky not to be rewarded.
The first unofficial Test ended in a high-scoring draw after notable double hundreds from Naman Ojha (219 not out in the first innings) and Marsh (211).
Brief Scores:
Australia A 1st innings: 288/7 in 90 overs (Peter Forrest 77, James Faulkner 94, Amit Mishra 4/114).
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