Kortney Hause headed the only goal in the 88th minute at Old Trafford, with Villa escaping with the 1-0 victory when Bruno Fernandes missed a stoppage time penalty.
It was just their second win over the Red Devils in 46 Premier League meetings and their first since 2009.
Smith said: “As everyone knows we have had a lot of suffering as Villa fans at this stadium and against Manchester United in general.
“People will look at it and say it is a big win but for me it was a big performance. I thought we deserved the win.
“I thought we had the bigger chances. I thought we were aggressive and brave in the way we played against a front four of Ronaldo, Fernandes, Pogba and Greenwood.
“In the first half I thought we were a little too passive at times. We allowed them to switch play probably one or two times more than I wanted.
“We felt we could get down the sides of them. What epitomised the gameplan was Matty Cash crossing for Matt Targett. That showed our aggression.”
Hause, making his first Premier League start of the season, looked like going from hero to villain when he handled Fernandes’ stoppage time cross to concede the spot-kick.
But the Portugal international blazed his effort over the bar to send 3,000 travelling supporters into rapture.
Smith continued: “It’s a big win. We have suffered here for many years as a club. I think the seeds of belief were sowed at Chelsea the other week.
“Although we didn’t get the result we took confidence from the way we played.
“I am pleased for Kortney. I thought he played ever so well coming in the other week and I felt he deserved that goal and that start.
“But also Tyrone (Mings). He got some stick after that mistake against Chelsea and we have just gone back-to-back clean sheets now.”