Brentford doubled their lead 10 minutes later, breaking from their own penalty area before Mads Rasmussen teed up Emiliano Marcondes for a first-time finish.
Toney nearly made it 3-0 moments later when his looping volley from outside the box hit the bar and bounced down on to the goalline.
Andre Ayew twice came close with headers for Swansea either side of half-time but their hopes of a comeback were reduced when Jay Fulton was sent off on 65 minutes for a foul on Mathias Jensen.
Brentford, who lost out to Fulham at this same stage last season, will become the 50th club to play in the Premier League, and the first to make their debut since Brighton & Hove Albion and Huddersfield Town did so in 2017/18.
The Bees join Championship winners Norwich City and runners-up Watford as the other promoted sides.
Brentford fans can now look forward to finding out who their side will play next season when the fixtures for the 2021/22 campaign are revealed at 09:00 BST on Wednesday 16 June.
The new season will kick off on 14 August and reach its conclusion on 22 May 2022.
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