Jack Nowell celebrated his return from injury with a late try, one of six scores for Eddie Jones's men, who can now look to a last-eight clash likely against either Wales or Australia.
The Pumas started the brighter at Tokyo Stadium, but Eddie Jones's men hit back quickly, a catch-and-drive from a line-out creating space out wide for May to dot down on the left-wing.
Passions boiled over in the 12th minute with an off-the-ball scrap, followed shortly by a red card from referee Nigel Owens for a high hit on England captain Owen Farrell by Tomas Lavanini.
The 14-man Pumas quickly found themselves stretched and fullback Elliot Daly exploited a huge gap on the left for a try that doubled England's advantage.
Scrum-half Ben Youngs scored another deep into extra time to give England a 15-3 lead at the break.
George Ford sealed the four-try bonus point early in the second half with further tries from replacement Nowell and Luke Cowan-Dickie – the last sparking another tousle.
Argentina had some success with 10 minutes to go, winger Matias Moroni finishing off a slick move with a try under the posts.
Teen Petaia shines in Wallabies win
Earlier on Saturday, teenage wing Jordan Petaia crashed over for a try on his debut as Australia rode out two yellow cards to beat Uruguay 45-10 and stay on course for the quarter-finals.
Australia's youngest ever World Cup player at 19 scored one try and made another, while Tevita Kuridrani and Dane Haylett-Petty bagged two each in the Wallabies' total haul of seven.
The result produced a vital bonus point for the Wallabies to move ahead of Fiji after three games in Pool D, where Six Nations champions Wales remain unbeaten.