Nissanka scored 100 not out off 52 balls with five sixes and 10 fours as Sri Lanka, chasing Australia's 181, reached 184-2 with two overs to spare in Kandy to win by eight wickets.
A victory by Zimbabwe over Ireland in Group B on Tuesday, or against Sri Lanka on Thursday, will eliminate the 2021 champions.
If Zimbabwe lose both, Australia will need to beat Oman in their final match on Friday by a big margin to get through on net run rate.
"We're in the lap of the Gods now I think," said Australia captain Mitchell Marsh.
"Lot of emotions in the room right now. Haven't been at our best. Disappointed bunch at the moment.
"We shouldn't have lost to Zimbabwe with all due respect to them and then we were under pressure.
"We will be watching the Ireland-Zimbabwe game. However, things are out of our control at the moment."
Nissanka was involved in a 97-run stand off 66 balls with Kusal Mendis for the second wicket with the wicketkeeper-batsman posting his third fifty in the competition.
Nissanka became the first Sri Lankan to score two hundreds in T20 internationals.
After Mendis's dismissal, Nissanka and Pavan Rathnayake put on 76 off 34 balls to see Sri Lanka to victory.
Nissanka played some glorious cover drives off the spinners and played some elegant flicks against the quicks.
- 'One of my dreams' -
"Very happy to score a hundred in a World Cup. It had been one of my dreams to be honest," Nissanka told reporters.
"Our fielding was outstanding today. We spoke about how being sharp on the field is important on a flat wicket like this.
"We knew they were 20 runs short. Here the par score is 200. The instruction was for me bat long even though the run rate kept climbing.
"We knew bad balls would come and it was important to cash in when they came along."
Sri Lanka came into the World Cup after a 3-0 drubbing at the hands of England, but have gained momentum and have won all three games so far.
"Happy to go through to the Super Eights,"said Nissanka.
"We had put in a lot of hard work but we didn't have much momentum leading up to the World Cup. Glad things are falling in line.
"Still, it's a long tournament and we need to keep fighting."
Australia's campaign has been plagued by injuries, with pace spearheads Pat Cummins and Josh Hazlewood out with injury.
Their captain Marsh, who missed the first two games, returned and gave them a rollicking start by scoring 54.
Marsh and Travis Head (56) scored 104 runs off 51 balls for the first wicket.
With Australia eyeing a total in excess of 200, Sri Lanka's slow bowlers kept them to 181 with leg-spinner Dushan Hemantha picking up 3-37.
But in going for quick runs Australia collapsed, losing their last six wickets for 21 in 24 balls.
