Villa are four points from safety with three matches to go after the victory - their first three points since January.
Speaking after the victory, Smith said: "We have found it hard this season and there is no getting away from that.
"I know these players can play under pressure. We have been playing tournament football since we came back.
"We haven't got the points we need but we're still in the tournament."
The 49-year-old coach has challenged his team to put "scoreboard pressure" on relegation rivals West Ham and Watford.
Smith explained: "The analogy I used with the players is today was this was our first semi-final. We can't control results elsewhere, just our performances.
"We needed to win it to get through to the second leg, if we go and win at Everton, another tough game, we can certainly put scoreboard pressure on the teams above us."
A brace from Egyptian winger Trezeguet gave Villa their first victory for 11 matches and Smith felt that the result was merited.
He said: "It was a much needed win. It's been a while coming. I thought our performance deserved it.
"They probably started better but after five minutes I thought we got into the game and created some opportunities.
"The last 10 minutes of the first-half I thought they had their best period and we probably scored against the run of play.
"The second half performance was really pleasing, we kept our foot on the gas and were aggressive and created a number of chances."