The Seagulls have now won their last three in the league, while Brentford, who remain 10th, have picked up just one victory from their last seven.
Anthony Knockaert had a super chance to send the home side ahead barely two minutes into the match after some sloppy defending from Brentford, but the midfielder was never really in full control and poked wide of the post from inside the box.
Knockaert, who impressed throughout the match, had another half-chance three minutes later, but the Frenchman's effort was deflected wide of the post.
Brighton were ahead in the 27th minute, however, when Knockaert worked a yard inside the box before finding the bottom corner with a low strike to open his account for the club.
The home side then had their second late in the first period when Tomer Hemed's well-placed header found the bottom corner after Beram Kayal had beaten Brentford midfielder Josh McEachran down the right for pace.
Knockaert came close to scoring his second of the night just past the hour after a low cross from James Wilson, but the midfielder's effort was over the crossbar.
Questionable goalkeeping from Stockdale presented Brentford with a chance to register 20 minutes from time, but John Swift saw his close-range effort blocked by Wilson.
Knockaert, who joined Brighton from Standard Liege in the January transfer window, had another golden chance five minutes from time when Jamie Murphy burst down the left before delivering, but the cross was just too strong.
There was time for one more Brighton goal, however, as Murphy burst through on goal in the third minute of additional time before finding the bottom corner to cap what was a miserable night for Brentford.