England posted their highest Murrayfield points total and scored four tries, with fly-half Paul Grayson kicking 15 points.
"This is a difficult place to come, and I thought that Scotland played very well," Woodward said.
"I am pleased to come away with a win, and we are now looking forward to taking the team back to Twickenham to play Ireland next month. It will take a good team to beat us there.
"We have been here in the past and lost, and the defeat we suffered here in 2000 was one of the worst I have experienced as a player or a coach.
"You have got to go through those experiences, and while we were a bit disappointed with the last 15 minutes, I felt that we won with something to spare."
Woodward paid tribute to Grayson, who again played with calmness and composure behind a dominant England pack.
"He has been outstanding," Woodward added. "We have got the number one fly-half in Wilkinson, but to have someone like Grayson available meant there was only one choice for the 6 Nations when Wilkinson was injured.
"He is not Jonny Wilkinson - I don't think anyone is - but I thought that he controlled the game very well.
"Overall, I thought that it was a good game, a good spectacle, and a credit to the 6 Nations."