After being restricted to a modest 8-203 from 50 overs after winning the toss and choosing to bat, a superb bowling and fielding display paved the way for victory at Lord's.
The win means Australia takes a two-point lead in the multi-format series after the opening clash, a sole Test match, was drawn.
The one-day world champions are chasing their first Ashes win on English soil since 2001 but they have plenty of work ahead with two more one-day games and three Twenty20 matches still to play.
QUICK SINGLE: Spinners are winners
"This gives a good momentum but it's a big series, there's still five games to go, and now we've got to focus on the next one," Australian captain Jodie Fields said.
"We would have liked to get a few more runs early on, it was a good batting wicket.
"We didn't quite play the shots we wanted to and read the conditions but that's cricket and we kept fighting today."
England made a solid start to their run chase before player-of-the-match Osborne (3-39) broke a 63-run opening partnership, with Sarah Coyte taking a great catch to dismiss opener Arran Brindle for 21.
Wickets remained hard to come by as England crept along steadily but the turning point came in the 38th over when Osborne had opener Charlotte Edwards (61) caught behind to halt the home side's momentum.
From there it was all Australia, with two crucial run-outs crushing England's hopes and Jonassen (4-38) and Coyte (1-20) cleaning up the tail.
"I think it was a lot of naive cricket from us," England captain Edwards said.
"A lot of credit has to go to Australia for holding their nerve but we took some poor options and it cost us.
"I don't think it's any time to panic. There's a lot of positives to come out of today and there's still a lot of cricket to be played."
Earlier, opener Meg Lanning top-scored for Australia with 56 on a fine day in London.
Australia's innings was off to a poor start as Rachael Haynes was caught and bowled by Brunt for a duck in the first over.
Lanning steadied the ship but an impressive bowling contribution from Laura Marsh (2-27) helped England limit the visitors to what looked a very gettable total.The series continues with the second one-day match in Hove on Friday.