The vast majority of the scoring came during the opening 40 minutes of the clash, with May's brace proving decisive come full time.
England, now unbeaten on home soil in 15 matches, were made to hold on in the final stages after Wales moved to within six points.
Before the late tension, though, it was a positive first-half showing from the hosts as May found the whitewash at the end of an Owen Farrell grubber-kick with just a couple of minutes on the clock.
England moved further clear midway through the first half when May was picked out by Joe Launchbury, and Farrell this time had better success from the follow-up.
The visitors registered their only points of the half three minutes later through Patchell, though they had a strong shout for a try rejected by the television match official when Gareth Anscombe appeared to get his hand to a chipped pass.
A tight second half ensued in the English capital, with Wales squandering their best chance of bagging the try they desperately required as Scott Williams failed to cross the line from close range.
Instead, replacement Sam Underhill made up ground to drag him out of play agonisingly close to the try-line, proving to be a possible match-winning challenge.
Wales's only other points would come four minutes from the end when Anscombe booted the ball down the middle to bring his side within a converted try of topping the hosts, but Eddie Jones's men saw things through to make it two wins from two in this year's competition.