Australia stumbled to 213-7 at lunch on day three as Smith chiselled England's lead down to 89 runs.
Steve Smith was summoned to the middle after just 10.3 overs following the cheap dismissals of Cameron Bancroft (5) and Usman Khawaja (11).
Cummins dismissed Stoneman, with England reaching 128/2 at tea on day one of the Ashes opener.
Starc struck in the second over but England manged to reach 59/1 at lunch at the Gabba in Brisbane.
A brilliant dismissal on the ropes during the eight-over match couldn't save the Kiwis in the end.
Hardik Pandya flew out of nowhere to snare the opener as the Black Caps lost by 53 runs.
The Black Caps opener set the tempo for chasing the ODI target of 337 with a knock of 75 off 62 balls.
Pandurang Salgaonkar allegedly told reporters posing as bookies the wicket was a hitter's pitch.
Brad Schmulian broke George Watson's record for most runs on debut with a knock of 203.
The Kiwi skipper was sent packing for just three runs by Jasprit Bumrah and wasted the Black Caps' only appeal in the process.