"We didn't do it for the full 80 minutes but we are very excited about what we are doing and this is a step in the right direction for us. It's about possession and taking control."
Care was quick to highlight the significance of Jones's yellow card as England avenged defeats to the Welsh in their previous three Six Nations encounters.
"Obviously these sort of games come down to little errors," Care told BBC Sport.
"He stuck out a leg and he had to go for 10 minutes which gave us a big lift and it was great to get that try before half-time."
Centre Mathew Tait, whose deft pass out of the back of his hand sent Haskell in for his second try, added: "When a team goes down to 14 men you've got to capitalise on that and I think Steve Borthwick and Jonny Wilkinson guided us well through that period."
England will now turn their attentions to facing Italy, who lost 29-11 in Ireland, in Rome next Sunday.
"You see how much hard work everyone puts in but sometimes it doesn't come to fruition on the field," said captain Borthwick.
"I'm very proud of the guys. We work hard and sometimes it doesn't come to fruition but today it did. We've been a very tight unit and we are all learning from Martin Johnson. We must do better and that is what our focus will be going into next week."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.