Ben Stokes (70), Ian Bell (55) and Alastair Cook (44) laid the platform for the innings, before Jos Buttler rammed home the advantage with a dazzling 71 off 43 balls.
Buttler's knock featured six fours and four sixes as Australia's pacemen were carted to all parts of the ground.
Johnson finished with 0-72 from his 10 overs, while fellow speedster James Pattinson (0-63 off eight) also struggled.
Allrounder James Faulkner finished with 4-67 after claiming wickets off consecutive balls in the final over as England tried to hit out.
England trail 3-0 in the series, and are desperate to snap a nine-game losing run that is just one shy of the record they set in 1993 and 2001.
Australia entered the match missing a host of their best players, with skipper Michael Clarke, wicketkeeper Brad Haddin, allrounder Shane Watson and opener David Warner all rested.
Stand-in skipper George Bailey elected to bowl after winning the toss, but the decision backfired as England raced to 0-73 from their first 10 overs.
Before the match, Cook revealed he wanted to stay on as one-day skipper despite his team's disastrous tour of Australia.
The 29-year-old hit a series of glorious strokes through the off side before he was bowled by spinner Glenn Maxwell (1-37 off nine) attempting an ambitious slog sweep.
Shaun Marsh was looking for a hole to hide in after dropping Bell and Stokes in the same over.
Bell was on 52 when Marsh dropped a sharp chance above his head at first slip.
Three balls later, Stokes received a life on 30 when a diving Marsh couldn't hold on to what would have been a spectacular one-handed catch.
Marsh finally managed to get one to stick when Stokes played Pattinson to mid-wicket.
However, Marsh was uncertain whether the catch had carried, and Stokes was allowed to bat on after replays proved inconclusive.
Stokes, who scored his maiden Test century in Perth in December, entered the match with an average of just 13.66 from his 13 ODIs.
But he made the most of his double slice of luck to lift his average to 19.3 by the end of his knock.
Buttler and Eoin Morgan (33) combined for a 71-run stand off 51 balls near the end of the innings to help lift England above 300.