He was left needing five off the last ball but could scramble only two from a well-placed yorker as England celebrated an unlikely victory that puts them one-up in the three-match series.
We probably struggled to find the boundary in that 12 to 18-over markAustralia captain Finch
“The bowlers really came good in the last eight overs. It's always difficult coming in, under lights, pressure on, on a slow wicket,” said England captain Eoin Morgan.
“Today we went with a batter less, which proved very valuable in the second innings.”
England, fresh from their series win over Pakistan, started strongly as opener Jos Buttler hammered 44 off 29 balls.
Dawid Malan then continued his remarkably consistent form with a superb 66, made all the more challenging as nobody was able to hang around long to build a partnership in what looked a below-par total.
Australia showed no signs of rustiness in their first game since March as a brilliant 98-run opening stand by David Warner (58) and Aaron Finch (46) put them in command.
Steve Smith then hit a quick 18 but after he fell to make it 124-2 Australia slumped to 148-6, losing four wickets for nine runs in 14 balls.
Ashton Agar was run out by an inch on the last ball of the penultimate over, leaving Stoinis to come up just short in the roller-coaster final over.
“We probably struggled to find the boundary in that 12 to 18-over mark,” said Australia captain Finch.
“We had a couple of great hit-outs against each other, but once you play England, it's a bit different.”