England won the fourth Test here Wednesday by an innings and 157 runs to retain the urn on the Australian soil after 24 years.
Strauss said winning away has always been tough and he is "immensely satisfied" with the team's performance.
"Winning the Ashes in Australia has always been a bit of a holy grail for English sides," Strauss was quoted as saying by the Guardian.
"We have not won the series yet, but have retained the urn - which was one of our primary goals. The players deserve everything they get, because they have stood up when it matters. The back-room staff have prepared us well, and we sit here today feeling like a lot of hard work has paid dividends for us."
Strauss felt the outcome of the Melbourne Test was decided on the first day when the home team suffered a batting collapse.
"The pressure of the situation - a Boxing Day Test, massive Test match, huge atmosphere, Ashes on the line - for the bowlers to perform the way they did on the first morning was outstanding."
"Trotty then stood up with the bat, because we realised this was our chance to hammer it home. From that position, it was always going to be hard not to win the game," he said.
England lead the series 2-1 and Strauss picked both victories in Melbourne and Adelaide as the best of his career.
"Two of the Tests we have played here have been two of the best I have been involved with," he said.
"We have played some outstanding cricket - obviously Perth was a bit of a comeback down to earth and a reminder that we are not the finished article yet."