Wales had struggled in the first half of their first two games, against France and Italy, but once again they dragged themselves into a contest they had struggled to dictate.
Anscombe banged over a penalty on the angle after May had been penalised for holding on and the margin was down to four points with more than a third of the contest to come.
Sinckler had played on the edge throughout but was penalised in the Welsh half for obstructing Anscombe and then again for a high tackle on Alun Wyn Jones in front of his own posts.
As the England tight-head was hauled off for Harry Williams, Anscombe knocked over the penalty for 10-9 and the home support was alive again.
It was Anscombe's last act, Biggar thrown on with 20 minutes left, but when Curry won a penalty as Parkes held on 25m out it was his opposite number Farrell with the three points.
Back came Wales. Drive upon drive hammered at the English defence, the big ball-carriers taking it up to within a few metres, the crowd baying as the red shirts got closer.
On the 35th phase the white line cracked. Hill battered through two men on a clever angle and reached out to nudge the ball on to the try-line, and Biggar's sweet kick from the right touchline made it 16-13.
And with England pinned back Adams soared above Elliot Daly to take the ball at the second attempt and flop into the corner.