Targett and Trezeguet were on target as Aston Villa beat Leicester City to set up a Wembley final against Manchester City or Manchester United.
Targett scored in the 12th minute to out the hosts ahead in the match but Leicester’s Kelechi Iheanacho’s 72nd minute equaliser restored parity.
Having played a 1-1 draw away at the King Power stadium, the game was set to drag to extra time and possibly penalties at the Villa Park before substitute Ahmed Elmohamedy sent over a superb cross for Trezeguet to drill a far-post finish across Leicester keeper Kasper Schmeichel.
It sparked scenes of celebration and a pitch invasion as Villa was reaching their first silverware showpiece since they lost in the FA Cup final 4-0 to Arsenal.
This was also the first time Villa will be in the Carabao Cup final since 2010 when they lost to Manchester United.
Speaking after the match to Sky Sport, Aston Villa manager Dean Smith said:
“I certainly let myself go with the goal. That deep into stoppage time, there was a bit of emotion from everyone and the place erupted.
“Our goalkeeper made three great saves but I thought we were aggressive as well.
“To go and get the win at the death like we did there, to take this club to Wembley, I’ll be a very proud man to walk out there with this team.”
Leicester manager Brendan Rodgers told Sky Sport:
“I thought we were excellent. Their goalkeeper made some outstanding saves.
“We had a nice control in the game. I thought we should have had a definite handball for a penalty. By the end of the game I thought only one team looked like they were going to win it.
“It’s tough to take looking back over the two games, but congratulations to Aston Villa and we have to move on.”
The Man of the Match was Villa’s goalkeeper, Orjan Nyland.