Australia sealed a dominant victory before lunch on Monday, having chased 170 runs without the loss of a single wicket in a blow to England's pride.
“We had to show some really good character throughout," Smith said at the post-match presentation.
"Obviously losing the toss, and the wicket was a lot slower than we anticipated it would be. We had to work really hard.
"I thought the bowlers did a terrific job to get them out for 300."
Smith singled out spinner Nathan Lyon for his contribution to the win.
"Nathan Lyon was particularly good. He is going from strength to strength and bowling like a genius.
"He did a terrific job holding up one end. The others got the wickets."
Smith almost single-handedly pulled his team out of danger with an unbeaten 141 on day three that ensured Australia could mount a charge towards victory in the second innings.
"Hard work at four for 70," he said. "Had to dig deep and get a big total and put on a few good partnerships. We got ourselves in the lead, bowled well in the second innings, the two boys did a terrific job to get us home."
England captain Joe Root said Smith had proved the difference and lamented his batsmen's failure to put a big score on the board.
"For three days we were excellent. Bar Steve's knock we were right in it," he said at the presentation.
"At 250-odd for four (we were) looking at trying to get 400 plus and put pressure on the scoreboard.
"In that position, having an 80-run lead would have been massive. It's frustrating. We have to move on quickly and make sure we don't make the same mistakes in Adelaide."
The second test, the first day-night Ashes clash, starts in Adelaide on Saturday.
First Published: November 27, 2017, 9:25 AM IST