Southampton’s first win in five matches moves them 10 points clear of the relegation zone but Villa, who managed just one shot on target, remain a point above the bottom three.
They have also suffered three straight defeats as they prepare to face EFL Cup holders Manchester City at Wembley.
“My feelings are embarrassment with the performance,” said Smith. “The thing that hurts me the most is a lack of fight.
“They’ve let themselves down with their own personal performances. We got beaten and deservedly so.
“Our quality just wasn’t there. Some of them have played their way out of a cup final.”
