Leon Knight scored twice in seven first-half minutes to seal Swansea's place in the League One play-off final.
Knight's goals were the difference between the two teams
Knight lifted the ball over Stuart Nelson from 20 yards for his first and then raced on to Leon Britton's pass before drilling home the second.
Rory Fallon missed an excellent chance to extend the Swans lead, heading wide.
Swans keeper Willy Gueret saved a curling free-kick from Kevin O'Connor and two long-distance shots from Isaiah Rankin as Brentford rallied.
Swansea had not won at Griffin Park in their previous 14 visits, a run that stretched back to 1977.
But Kenny Jackett's side, who were held to a 1-1 draw in the opening leg of the tie, will now make their second trip to the Millennium Stadium this season after winning the Football League trophy in April.
The result brings to an end a disappointing finish to the campaign for Brentford, who narrowly missed out on automatic promotion after drawing six of their last seven games of the regular season.
Brentford, who lost to Sheffield Wednesday at the same stage last year, have now been unsuccessful in six end-of-season play-off campaigns.
It might have been very different for Brentford if Jamie Smith had converted a good chance after just 20 seconds.
The ball broke to Smith on the edge of the area but he made a poor contact and his shot was comfortably saved.
A series of crunching tackles followed as both teams looked to take control - but Knight's brace soon ensured that Swansea were firmly in command.
The opener came after Fallon got the slightest of flick-ons to Willy Gueret's long ball forward and Knight then hooked the ball over Nelson with his outstretched right leg.
The second, with 15 minutes gone, was a brilliant finish - Knight smashing the ball past Nelson with his second touch after controlling Leon Britton's excellent through ball.
Bees boss Martin Allen brought on Callum Willock for Ricky Newman after 24 minutes as he looked to manufacture a change of fortune.
Jay Tabb shot wide and Alex Rhodes forced Gueret into action but the Swans continued to pose the greater goal threat and Fallon should have extended their lead before the break only to head wide when completely unmarked.
Allen made a further two substitutions at half-time, introducing Paul Brooker and Sam Tillen - and his team enjoyed a sustained spell of pressure as they tried to force their way back into the tie.
Gueret saved a free-kick from Kevin O'Connor and a strike from Rankin, while Willock headed over from a Brooker cross.
Rankin forced another save from Gueret late on before Swansea almost extended their lead after hitting Brentford on the break but Tom Williams could not keep his effort on target.
Brentford manager Martin Allen:
"We have a lot of good players out of contract, and we go from this game into making decisions about players' careers.
"I will settle down on Monday, meet my staff and make decisions about players.
"I've no complaints about the result. Swansea were stronger than us and Leon Knight produced two quality finishes."
Swansea manager Kenny Jackett:
"It was a battling performance. Griffin Park is a difficult place to come to and we handled it well.
"We have withstood the pressure, there were holes to break into and we could have done better in the second half.
"But I cannot complain and I am now looking forward to the final - I am delighted to be there."
Brentford: Nelson, O'Connor, Sodje, Turner, Frampton (Tillen 45), Smith (Brooker 45), Newman (Willock 23), Pratley, Tabb, Rhodes, Rankin.
Subs Not Used: Bankole, Hutchinson.
Swansea: Gueret, Tate, Austin, Monk, Ricketts, Britton, O'Leary, Tudur-Jones, Robinson (Williams 74), Knight (Forbes 67), Fallon (Trundle 85).
Subs Not Used: Murphy, Akinfenwa.
Booked: Robinson, Fallon.
Goals: Knight 8, 15.
Ref: G Laws (Tyne & Wear).