Chris Wood had fired the hosts in front after just four minutes, producing a fine first-time strike after Matt Lowton's pass evaded the outstretched leg of Bees defender Ethan Pinnock.
Right-back Lowton's header from Charlie Taylor's cross then doubled the advantage before Cornet ensured Burnley's first-half superiority was fully reflected.
Brentford were much-improved after the interval. Christian Norgaard forced Nick Pope into action and Ivan Toney went close after a rare moment of confusion in the Clarets' defence.
But while Saman Ghoddos' excellent volley eventually reduced Burnley's advantage with 11 minutes remaining, Frank's side were unable to produce a sensational late rally and slumped to a third successive league defeat.
He later said: "At half-time we talked about pride and getting back into the game. The second half was better.
"We win or learn. Today we definitely learned, especially in the first half. In the second half we grew into it, even though the dynamic changed a bit in the second half. We need to perform much better than we did in the first half."
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