The Australian felt that his players had given him the response he and the passionate Scottish crowd was looking for - with a second-half try - after the reverse in Wales.
"I'm really proud of the guys," said Williams. "I said before that, with the support you get from the people here, we had to be worthy of it and there were some fantastic signs out there.
"The defence was great against the great barrage of England pressure and there was a lot of spirit and effort.
"It was a huge performance from Scotland and we could have scored a couple more tries in the first half.
"We are a young team and we did make some mistakes with handling and in the line-outs but I'm proud of the players.
"We spent a lot of energy and effort stopping England, which is not easy, and I'm delighted that we've taken the first step. Right through the team they played very well."
Williams was disappointed with England's first two tries but he was surprised by how well his side responded.
He added: "Their heads could have gone down after those tries but they gave it everything and we can all be proud of them."