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The tournament hosts were staring at an early exit after crashing to 10 for three in reply to Sri Lanka's 122 for six.
The match was effectively a knockout contest for Australia given they lost their opener to India.
Haynes (60 off 47 balls) and captain Meg Lanning (41 not out) guided Australia to a five-wicket victory with three balls to spare.
Australia need to beat Bangladesh in Canberra on Thursday and New Zealand in Melbourne on March 2 in order to have a chance of making the semifinals.
Depending on other results, net run rate could decide which two teams progress from the group.
The pressure and expectation on Australia have been immense during the World Cup, and Haynes hopes the fighting win over Sri Lanka is the start of a winning run.
"Hopefully it just settles our group down a little bit," Haynes said.
"The other day [in the loss to India] it didn't go to plan, but that's the nature of tournament cricket - you don't always have it your own way.
"We knew we were going to get challenged at different times. Playing T20 cricket you are riding this wave of emotion and momentum.
"I think more than anything it will be really nice to carry that on heading into Canberra and that game against Bangladesh."
Australia showed great spirit against Sri Lanka, but their batting remains a concern.
The defending champions could only muster 115 against India, and their early wobbles against Sri Lanka showed some more cracks.
Haynes feels the team can continue to build from this point.
"I hope [the win over Sri Lanka] showed a bit of our character," Haynes said.
"To be able to get over the line right at the end there, it's going to hopefully create some wonderful momentum heading into the rest of the tournament."