Nissanka, who finished with exactly 100 runs, led the way, and his second-wicket partnership of 97 runs with Kusal Mendis laid the foundations for Sri Lanka's third straight Group B win.
Australia, former champions who lost their last match to Zimbabwe, must hope Ireland and Sri Lanka beat Zimbabwe to have a shot at qualifying for the Super Eights with a top-two finish in the group before they play Oman on Friday.
"We are in the lap of the Gods now. Lot of emotions in the room. We have not been at our best. We watch the Zimbabwe-Ireland game with hope, but we are disappointed," Australia captain Mitchell Marsh said.
Put in to bat first in Kandy, captain Marsh and Travis Head (56) got off to a fast start with a 104-run opening partnership inside nine overs.
Head completed his fifty in 27 balls, shortly before getting caught at wide long off looking to hit his fourth six, as Dushan Hemantha broke through for the co-hosts.
Hemantha (3-37) trapped Marsh (54) leg before wicket in his next over, and then dismissed Glenn Maxwell (22), whose reverse hit off a full toss was brilliantly caught by Nissanka with a backwards leap at backward point, leaving Australia at 160-5 in the 17th over.
Josh Inglis (27) was the next to fall to Dushmantha Chameera, who bowled a tight last over in which Australia scored only four runs and were all out after two run-outs in the last two balls.
Nissanka leads from front
Sri Lanka started well despite losing Kusal Perera early, as Nissanka and Mendis added 97 runs for the second wicket in 66 balls, until Mendis (51 off 38 balls) got caught trying to hit Marcus Stoinis (2-46) for a six.
With Sri Lanka needing 61 to win in the last six overs, Nissanka hit Stoinis for two fours and a six to score 20 runs off the over, easing the pressure.
The 27-year-old reached his second century in T20 internationals off 52 balls, hitting five sixes and 10 fours along the way, before Pavan Rathnayake (28) hit the winning runs with a sweep towards mid wicket, getting Sri Lanka to 184-2.
"One of the best performances in recent past. After many years, making it into the next round," said Sri Lanka captain Dasun Shanaka, referring to his team's exit from the group stage in 2024.
"Hopefully we can make it to the semis."