The Bees lost at this stage last season to London rival Fulham but were in no mood to be denied this time around and were 2-0 up inside 20 minutes.
Ivan Toney's 10th-minute penalty – his 33rd goal of an amazing campaign – opened the scoring and Brentford's lead was doubled by Emiliano Marcondes just less than 10 minutes later.
Swansea was much improved in the second half, but Jay Fulton's unfortunate red card effectively sealed victory for Brentford as it clinched a first promotion to the top flight since 1935, ending a 74-year absence from the highest domestic level having been relegated in 1947.
Toney had scored all 10 of his previous penalties this season and never looked likely to err after goalkeeper Freddie Woodman felled Bryan Mbeumo when he was played in by Sergi Canos' superb reverse pass.
He duly made it 11 from 11 by calmly slotting beyond former Newcastle United team-mate Woodman and into the bottom-left corner, with Toney's record from the spot the best of any player in England's top four tiers in all competitions.
Swansea struggled to respond to that early setback and was soon given a mountain to climb when Mads Rasmussen expertly picked out Marcondes at the end of a counter-attack and the Dane rifled a fine finish into the bottom-right corner.
The Bees were almost out of sight as Toney cracked a spectacular volley against the underside of the crossbar, with Woodman grateful to see the ball bounce on the line and out.
Swansea did stem the tide thereafter and went close in the final minute of normal time in the first half as Andre Ayew sent a header off the top of the bar before starting the second half by skewing a diving header wide from Connor Roberts's cross.
Jamal Lowe crafted some promising openings for the Swans but Fulton was shown a straight red card for a reckless challenge on Mathias Jensen, giving Brentford fans cause to start the celebrations early on a landmark day for the club.
Swansea was looking to end a three-season stay in the second tier following Premier League relegation in 2018, but Brentford will instead make its debut in the division, becoming the 50th different club to grace the league since its inception in 1992.
Its 86 years between promotions to the top flight is the second-longest gap behind Bradford City, which ended a wait of 91 years in 1999.
Brentford could start the season against the European champion, with games against Manchester City and Chelsea on the horizon.
For Swansea, another term in the Championship awaits, with the likes of Peterborough United and Luton Town lying in wait.