The promoted Bees scored the opening goal of the season when Sergi Canos smashed in a low 22nd-minute strike.
Christian Norgaard headed in Brentford's second as Arsenal made a mess of dealing with a long throw-in.
The Dane was then inches away from making it 3-0 after Gunners keeper Bernd Leno flapped at a corner.
Arsenal, without strike pair Alexandre Lacazette and Pierre-Emerick Aubameyang through illness, paid the price for a hopeless attempt to defend substitute Mads Sorensen's routine long throw. New £50m defender Ben White was particularly culpable as Christian Norgaard arrived to head home.
Brentford's supporters, understandably, celebrated wildly at the final whistle, but this was a result and performance that instantly turns the spotlight on Arsenal manager Mikel Arteta as he desperately needs an improvement on last season.
Brentford make powerful statement
Brentford could not have made a better start to life in the Premier League, showing the pedigree that has made them such an entertaining side on their journey to this stage.