A game lacking in too many clear-cut chances looked to be heading for a penalty shoot-out until Tabea Kemme brought down England substitute Lianne Sanderson.
Williams kept her nerve to send goalkeeper Nadine Angere the wrong way from the spot-kick, despite the attempts of Germany sub Alexandra Popp to put her off in the build-up.
Germany had an excellent chance to open the scoring in the first minute when Lena Petermann’s downward header was pushed around the post by England goalkeeper Karen Bardsley.
Lionesses skipper Steph Houghton was in the thick of the action at both ends, clearing off the line to prevent Jo Potter heading into her own net before scuffing an effort at the other after a good cross from Lucy Bronze.
The World Cup’s top scorer Celia Sasic then popped up in a good position but could only mis-hit a shot from 12 yards straight at Bardsley, with Sasic then seeing a volley blocked in the next attack in first-half stoppage time.
There was also a muted penalty appeal for England after a driven shot hit Kemme on the arm, but the referee waved it away.
Bardsley impressed in the England goal and the Manchester City stopper was called upon to save well from Sasic after she volleyed goalwards after getting on the end of a cross from Sara Daebritz.
Kemme caused problems as she ran confidently at the England defence in the second half only to slice her shot past the post.
England could have won the game in normal time but after sub Eniola Aluko had slipped the ball into her, Jill Scott dithered when she should have shot and the chance was gone.
Houghton was a rock at the back for Mark Sampson’s side and she positioned herself well to stop Lena Petermann converting Simone Laudehr’s cross in extra-time.
Germany also had the chance to secure victory just before the extra-time break, but Melanie Leupolz’s powerful 20-yard shot flew over the crossbar after they had worked the ball around well.
Williams then calmly slotted home the winning penalty after 108 minutes of play but there was still time for some late drama, with Bianca Schmidt failing to connect properly with a back-post header to cue England celebrations soon afterwards.