Byrne identified his colleagues in the Ireland pack as the key to a victory which got Brian O'Driscoll's team back on track following last week's defeat by France.
"I thought the forwards were absolutely fantastic today," he told BBC television.
"We knew we were going to be up against a very strong pack and would have to take it to them. We did, and it was fantastic."
Byrne and O'Driscoll are stepping into the famous boots of Keith Wood, the former as hooker and latter as Ireland's new captain.
They each marked this convincing performance with two of Ireland's six tries, and Byrne's only regret was that the whole experience did not last a little longer.
"This was very important for the confidence, because I do not think we did ourselves justice in France," he said.
"Everybody was absolutely tuned in and fully committed to the cause. It was a great day, and I just did not want it to end.
"It is a dream come true."