Two goals inside 11 first-half minutes saw the hosts lead at the break, but Dagenham hit back.
Sam Saunders pulled one back for the visitors, who then piled on the pressure after Kevin O'Connor was harshly sent off with 13 minutes remaining.
But Brentford held firm and extended their own unbeaten run to five games.
After an even opening, Brentford took control midway through the first half when Nathan Elder headed home an Adam Newton cross to give them the lead.
The visitors had clipped the post early on through Ben Strevens, while Saunders also hit the woodwork with a free-kick, but they couldn't deal with a rampant Brentford at the end of the first half.
Charlie MacDonald doubled Brentford's lead 11 minutes before the break when he finished off a quick break away by slotting a Glenn Poole pass under Tony Roberts.
The Dagenham keeper had already smothered at the feet of MacDonald and tipped a rasping shot by the same player onto the post as he fought to keep his side in it.
But the game changed after half-time when Saunders curled home from the edge of the box and Dagenham pressed.
Brentford goalkeeper Ben Hamer had to be alert to save from Saunders and also denied Richard Graham.
Roberts saved well from Poole at the other end and Elder headed over, but the game swung in Dagenham's favour after O'Connor's dismissal for a lunging tackle on Strevens.
But the visitors couldn't break down a stubborn Brentford defence and slipped off the top of the table.