Ollie Watkins’ composed finish in the 11th minute levelled the semi-final on aggregate, and a header from Emiliano Marcondes put the Bees ahead in the tie four minutes later.
Bryan Mbeumo volleyed in a cross from Rico Henry in the first minute of the second half to make it 3-0 on the night and give Brentford a two-goal cushion.
Rhian Brewster dragged Swansea back into the contest with 12 minutes remaining, capitalizing on a poor clearance from Pontus Jansson and chipping home goalkeeper David Raya, but the visitors were unable to find a second goal which would have forced extra-time.
Victory meant the Bees signed off at Griffin Park, their home of 116 years, in style before their move to a new stadium this summer.
The west Londoners, who are looking to end their 73-year exile from the top flight, will face either Cardiff City or neighbors Fulham at Wembley on Tuesday, 4 August.
Fulham will take a 2-0 lead into the second leg of their semi-final tie on Thursday
Reaction
Brentford head coach Thomas Frank: “We talked about not leaving the pitch without getting to the final. The message was we needed to start faster than we ever started before. And we did that.
“At 3-0 we should have been through to the final, but we made a mistake we shouldn’t make. But after that they weren’t really dangerous and it was nice to show the defensive side of our game.
“I am pleased and happy. It is a very nice moment and we have achieved something good. But we want to achieve something big and we are going to the final to win.”
Swansea head coach Steve Cooper: “We gave ourselves too much to do. We just didn’t play well enough in the first half. We weren’t aggressive with our play, both with and without the ball.
“We ended up being too deep, leaving too much space around our midfield line and our defensive line and allowed them opportunities to attack.
“The goals are obviously really disappointing. The first one is from our set-piece and the second and third ones, although they are really good finishes, you cannot leave people unmarked in the box like that.
“We didn’t give up and we went right until the very end but we gave ourselves too much to do.”
