The win in Lucknow gives Australia some room to move after they dropped their first two matches and risked crashing out of contention for the knockout stage with another loss.
Things didn't always come easily for Australia, who lost the toss and struggled to contain Kusal Perera (78) and Pathum Nissanka (61) during a 125-run opening partnership.
Chasing 210 after bowling Sri Lanka out, Australia's recent batting issues threatened to resurface when David Warner (11) and Steve Smith (0) fell in the same Dilshan Madushanka over.
But ultimately, the 9-52 Sri Lankan collapse triggered by Zampa (4-47) and Pat Cummins (2-32), followed by half-centuries for Josh Inglis - a career-best ODI knock of 58 from 59 balls, and Mitchell Marsh, separated the sides.
Josh Inglis scored an ODI career-best 58 from 59 balls to help guide Australia past Sri Lanka's 209.
Once victory was in their grasp Australia, seeking to boost their net run rate, accelerated to reach the target in the 36th over.
Criticised for his form to begin the tournament, Zampa said he still had room for improvement once he returned to peak physical condition.
"I've been sporting a bit of a back spasm for the last couple of days so I tried to get through this one," said Zampa, who received treatment from physiotherapists throughout the innings in the field.
"I feel like I'm a good critic of myself and I feel like I could be doing better but it's nice to be on the better end of the result tonight."
Pat Cummins bowling Sri Lanka's Kusal Perera for 78 proved a key wicket for Australia.
Sri Lanka have now lost their first three matches and face an uphill battle to make it past the pool stage.
Makeshift opening batter Mitch Marsh (52 from 51 balls) recaptured his pre-World Cup form on a proactive start to Australia's innings at the crease.
He was resolute as Warner and Smith both went lbw, the former furious at falling victim to a DRS review that deferred to the umpire's on-field call of out.
Inglis then began a mature innings after Marsh was run out by Chamika Karunaratne, recognising patience would win the day for Australia given the low target.
He continued to stake his claim to remain in the XI ahead of usual wicketkeeper Alex Carey but took a back seat when Glenn Maxwell (31 runs from 21 balls) came on for a furious cameo late.
By the time Inglis was caught at backward point from Dunith Wellalage's bowling, Australia needed only 18.
Earlier, Cummins triggered a collapse by taking the breakthrough wicket of Nissanka, one of two Sri Lankan batters caught in Warner's fielding masterclass. He then sent Perera packing with a length ball that careered into off-stump for 2-157 .
The captain later punished Dunith Wellalage for attempting to sneak a single in by hitting the stumps directly from mid off.
Zampa finished the day with his best figures in an ODI World Cup match. The leg-spinner dispatched Sri Lanka's two most in-form batters, Kusal Mendis (9) and Sadeera Samarawickrama (8), in the space of two balls, the first falling to a fine diving catch from Warner.
David Warner took a spectacular catch in the deep to dismiss Kusal Medis off Adam Zampa.
"It took me a little while to get into the game today, I didn't feel at my best again but I've just got to try and keep that wicket-taking attitude," Zampa said.
Some wild wind in Lucknow blew banners down from the roof of the stadium and into the crowd, delaying the start of the second innings as ground officials secured the area.
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