Goals from Conor Hourihane and Anwar El Ghazi in the space of four first-half minutes eased Villa's recent woes after a run of three straight defeats had plunged Smith's side to one place above the relegation zone.
Inspired by returning captain Jack Grealish after a month-long absence with a calf injury, Villa ensured it was an unhappy visit back to Villa Park for former boss Steve Bruce, who was sacked in October last year after two years in charge.
Grealish, sidelined for the 2-1 defeats at home to Liverpool and at Wolves, again provided the spark Villa had been missing in those games.
The midfielder was fouled in the build up to the opening Villa goal on 32 minutes, when he was bundled over by DeAndre Yedlin on the edge of the area and Hourihane stepped up to whip the ball into the back of the net.
Grealish was also involved in the build up to the next four minutes later, as was Hourinhane, who provided the cross for El Ghazi to finish from close range.
The three points came as a big boost to Villa, who can now breathe a little easier, and Smith admitted his relief after the game.
"This is a really tough league and we have just gone through a period of playing Manchester City and Liverpool, which were games I thought we actually played well in," he said.
"The table is so tight but you need to put a run together, be tough to beat and keep a clean sheet. I'm really pleased with the clean sheet as we haven't had too many of those this season.
"Our form has been good; we knew it would be hard, but I think we've been competitive. It takes time to gel defensively but we achieved that against Newcastle."
Villa are next in action when they travel to Old Trafford to take on Manchester United on Sunday.