The game at Villa Park looked destined for extra time after Kelechi Iheanacho cancelled out Matt Targett's opener to make it 1-1 and 2-2 on aggregate.
However, Trezeguet steered home a right-wing cross from Ahmed El Mohamady on the volley in stoppage time to spark scenes of jubilation, with fans invading the pitch at the final whistle.
Villa, which last won the trophy in 1996 and was runner-up in 2010, is likely to face Manchester City in the showpiece, with Pep Guardiola's side holding a 3-1 aggregate lead over Manchester United ahead of the second leg at the Etihad Stadium.
"I certainly let myself go with the goal. That deep onto stoppage time, there was a bit of emotion from everybody when that goal went in and the place erupted," Smith said.
"To go and get the win at the death like we did there – we had it last week against Watford in the last minute and we've had it again today.
"But to be taking this club to Wembley, we've got a great history and a proud history in this competition, and I'll be a very proud man leading them out at Wembley."
With Pepe Reina missing out due to a tight calf, goalkeeper Orjan Nyland featured for the first time since conceding six against City on 12 January, making five saves across the 90 minutes.
"Our goalkeeper made three great saves early, two low down and one tipped onto the crossbar," Smith said. "But I thought we were aggressive as well and it was a good game of football with both teams trying to win it.
"We certainly are [making a habit of late wins] and it's certainly a good time to score, because you can see from the eruption in the stadium that they know it's very difficult to get back in it.
"I'm proud because the players were aggressive, had chances and created chances and made it a really tight game."
Leicester manager Brendan Rodgers felt his team deserved to win and was disappointed they did not get a first-half penalty when James Maddison's shot struck Marvelous Nakamba's arm.
"I thought we were excellent in the game. Their goalkeeper made some outstanding saves. We played with a nice control and composure in the game, created opportunities," Rodgers said.
"I thought we had a clear handball, it looked a definite penalty, so I'm just gutted for my players because they put so much into the game. Especially coming from behind, playing with that composure and scoring a very good goal.
"By the end of the game I thought there was only one team looked like they were going to win it, but we switch off from the cross and the guy gets in at the back post and scores it.
"It's tough to take looking back over the two games but congratulations to Aston Villa, they go through and we have to move on."