Swansea City led the tie from the first leg 1-0, but found themselves behind to an Ollie Watkins goal after just 11 minutes.
Four minutes later Emiliano Marcondes scored what looked to be the the winner, putting the home side two up on the night.
Bryan Mbuemo made it three straight after the restart and Brentford appeared to be cruising until Rhian Brewster made for a nervy finish with a 78th minute strike.
The Bees held on for a 3-2 win on aggregate.
Speaking after the game to Sky Sports, Brentford defender Rico Henry said:
"We had chances in the last couple of games but haven't been putting them away. Today we put them away and got the win.
"The last game at Griffin Park was always going to be an emotional one.
"We'll go to Wembley full of confidence whoever it is and we'll be looking to win."
Goalscorer Watkins was pleased to have improved their form at a vital moment: "I'm over the moon. We've had a tough few last games of the season and it hurts a little bit but we bounced back tonight.
"We came off the pitch angry and ready to go again. We started the game well tonight and we deserved it. We could easily have punished them a little bit more tonight."
Brentford boss Thomas Frank was full of praise for his players, but thought the scoreline might have been flattering to their defeated opponents.
"We were just so on it. I honestly think we should've won 3-0 or 4-0, then of course we make the mistake, which is football - and why we love it," he said.
"I said after training yesterday that we wouldn't leave this pitch until we're in the final.
"I felt we were the better team and thought we had that irritated feeling in our bodies."
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