Watson made it five British players into round two in Melbourne for the first time since 1987 after Kyle Edmund and Johanna Konta also progressed.
Naomi Broady nearly made it six but was edged out by Australian 22nd seed Daria Gavrilova, calling the crowd “pretty harsh” with their bias towards the home player.
Andy Murray and Dan Evans had already won on Monday.
Stosur, seeded 20th here in Melbourne and a local favourite despite having a dismal record at her home tournament, is the highest-ranked opponent, at 21st, that Watson has beaten at a grand slam.
In 17 attempts against players in the top 50, this was just her second success, the other coming in a mammoth match against Frenchwoman Caroline Garcia at Wimbledon in 2015.
Watson’s 6-3 3-6 6-0 victory also marks a dramatic upturn in form, given she failed to go past the second round of any tournament from June last year and watched her ranking drop from 50th to 81st.
Asked for the cause of the difference, Watson said: “Because I’m fresh. I haven’t played matches in a while so I’m motivated. I’ve had that time off and been at home.
“When it gets hard for me is when I’ve done too many weeks on the road and all I think about is going home.
“You have those weeks sometimes and you have to tough through them but it’s not good when it becomes every single week. I just want to manage that better.
“I need just to listen to my body. I think the Olympics in Rio was a big part of it last year. I talked about it a lot, was really looking forward to it, wanted to make it.
“Then I made it and once it was done I was like: ‘I just want to go home’. I was dead.”
Konta could not deliver the same brilliance that won her the title in Sydney last week but still proved too strong for Kirsten Flipkens.
The British No 1 produced a scratchy performance in her opening match on Margaret Court Arena but it is a sign of her rising class that she could still convincingly overcome Flipkens, under-ranked at 70th in the world, 7-5 6-2.
The ninth seed will face Japan’s big-hitting but erratic 19-year-old Naomi Osaka, who needed three sets to overcome Thai wildcard Luksika Kumkhum 6-7 (2/7) 6-4 7-5.
Konta’s breathtaking wins against Eugenie Bouchard and Agnieszka Radwanska at the Sydney International last week have many tipping her as a contender for the title here but this was not a vintage display from the 25-year-old.
Broady provided more evidence of her excellent progress in the day’s final match on Margaret Court Arena but lost out to the Russian-born Australian Gavrilova 3-6 6-4 7-5.
“It helped ‘Dasha’ a lot having the crowd behind her – when it’s such a close match, that makes a big difference,” said Broady.
“They were pretty harsh the longer the match went on. They started shouting between first and second serves but I’ve had way worse so I was prepared for it.”