The Lionesses gained some consolation for their heartbreaking loss to Japan in the semi-final as they produced a gritty display to beat the twice world champions for the first time.
A fiercely competitive game, scoreless after 90 minutes, was decided three minutes into the second period of extra time when England sub Lianne Sanderson was pulled down in the box by Tabea Kemme and Williams coolly slotted home.
The third place finish, behind United States and Japan who meet in the final on Sunday, is the best by an England team at a women's World Cup.
It also marks the best finish by an England team of either gender since the men's team won the World Cup in 1966.
"These players are just incredible. Now it's time for a massive celebration," said England coach Mark Sampson.
Germany, who had won 18 of the previous 20 meetings between the teams, started strongly and forced England keeper Karen Bardsley into action early with a powerful first minute header from forward Lena Peterman.
An injury time own goal from Japan had doomed England in the semi-final and there was almost more self-inflicted damage when a header from Jo Potter flew past Bardsley only for skipper Steph Houghton to produce a brilliant clearance on the line.
Bardsley was outstanding again in the 53rd minute to keep out a superb volley from Sara Dabritz and then Kemme shot just wide after a powerful run in from the left flank.
England finished strongly, though, camping deep in the German half and keeper Nadine Angerer did well to save bravely at the feet of substitute Eniola Aluko.
Germany dominated the first period of extra-time but the Lionesses found a second wind and the decisive penalty was reward for their pressure.
Germany coach Silvia Neid, who will stand down next year, said she had no complaints about the penalty or the result.
"We fought, we worked very well and had chances but if you don't score you aren't going to win. The penalty was justified, we were very naive. It is sad but it is reality but England won and deserved to win," she said. - Reuters
