After losing the toss, Sri Lanka A were in trouble at 5-89 at Marrara Stadium on Sunday before Dinusha steadied the ship.
The 24-year-old left-hander hit 10 fours and a six on his way to 105 not out from 148 balls before he ran out of partners.
Dinusha had help from Vishad Randika (27) and Isitha Wijesundara (33), while No.3 Nuwanidu Fernando (34) also made a start.
Jake Weatherald and Campbell Kellaway are set to open for Australia A when they start their reply on day two.
Right-arm quicks Liam Scott (2-27) and Henry Thornton (2-31) each took two wickets for Australia A, as did Nathan McSweeney (2-51) with his off-spin.
Mitchell Perry, Louis Smith, Zanden Jeh and Kellaway also chipped in with one wicket apiece as the hosts spread the load.
Jeh, a 21-year-old form Brisbane who has never played top-level state cricket, took 1-59 from 11 overs on his first-class debut, as Australia's search for left-arm spinners continues.
"I was a bit nervous, to be honest," Jeh said.
"I felt I started all right. I probably lost my way a little through the middle there, and when you do that against quality batsmen they'll put you away.
"If that's the case, it is what it is, but I felt like I came back towards the end there and finished up the day not too bad."
Jason Sangha bowled just three overs and was the only one of the eight Australia A bowlers used not to claim a wicket.
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