May crossed twice in the space of four first-half minutes to give England, which was humiliatingly eliminated on home soil four years ago courtesy of a pool-stage defeat to the Wallabies, a lead it never relinquished.
Australia was ultimately well beaten in what appears likely to be coach Michael Cheika's final game in charge, Kyle Sinckler and Anthony Watson also crossing in the second half as England, for which Tom Curry was outstanding, thundered clear.
Eddie Jones's men have now won seven successive Tests against Australia since that painful 2015 defeat and will face defending champion New Zealand in the last four.
The Wallabies had more of the ball throughout the contest, yet England was much more clinical as it showed no signs of rust in its first game for a fortnight, the 2003 champion's final pool game against France having been cancelled because of Typhoon Hagibis.
Teenage back Jordan Petaia — a bold selection from Cheika at outside centre — was prominent in a superb start for the underdogs, but the Wallabies' early dominance only yielded three points from Christian Lealiifano.
Having been initially pegged back by Australia's direct running and impressive ruck speed, England dramatically seized the initiative through May.
The Leicester wing's first score was a straightforward one as he accepted Curry's delayed pass to go over in the left corner, after Manu Tuilagi had played a key role in marching England forward.
A more eye-catching try quickly followed when Henry Slade intercepted a loose pass from David Pocock. Slade did not have the legs to reach the line, but he kicked ahead for May, who gathered calmly and outpaced Samu Kerevi to dot down again.
Owen Farrell, playing at five-eighth for the first time in the tournament, twice converted from the left touchline and added a simple penalty in between two three-pointers from Lealiifano, ensuring England led 17-9 at the interval.
Australia briefly reduced its deficit to a solitary point as Marika Koroibete streaked over from Petaia's inside pass, only for England to respond immediately, Farrell's flat cut-out pass laying on a try for Sinckler.
England never looked back thereafter and skipper Farrell kicked three further penalties before Watson's 76th-minute interception try rubbed salt in Australia's wounds.