The 29-year-old touched down late on as the Wallabies levelled the three-Test series 1-1 with a 16-15 triumph.
"It was really important to retain that belief we've been working on for so many years," Ashley-Cooper told Sky Sports News. "Six points down, there was a lot of frustration out there because possession was letting us down in terms of ball security.
"[The try] happened very quickly but I'm very proud to be a part of this team and score a try that made the difference tonight. It's not about me, it's about the team holding in there for 80 minutes and getting the result."
The Lions and the Aussies will play the deciding Test in Sydney next Saturday.