The hosts had dominated for the first hour, during which time English coach Stuart Lancaster's decision of selecting Owen Farrell ahead of George Ford at flyhalf looked inspired, with the 24-year-old kicking 20 points.
But sloppy, indisciplined play at the breakdown delivered the Welsh penalty after penalty, and they kept in touch thanks to the boot of Dan Biggar.
Wales scrumhalf Gareth Davies touched down after a superb Welsh move to level the match 25-25 with 10 minutes left and that man Biggar again struck with five minutes on the clock to hand Wales a famous victory.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
The win puts Wales top of Pool A with nine points, in prime position to make the knockout stages.
Both still need to face Australia, who play Uruguay on Sunday.
Only two teams will progress, meaning a nation currently ranked two, three or four in the world will fall at the first hurdle.
