Poole gave the Bees a 22nd minute lead and scored again in the second half after Ben Strevens had equalised for the home side.
Poole came close in the 15th minute when he fizzed a shot inches wide of Tony Roberts' goal from ten yards.
The striker made no mistake in the 22nd minute when he fired low past Roberts from the edge of the six-yard box to give his side the lead.
Daggers midfield Glen Southam shot straight at keeper Ben Hamer in the 33rd minute after dribbling through the Bees' defence.
Gary Smith should have doubled Brentford's lead in the 36th minute but pulled his shot wide from point-blank range after slipping behind the Daggers defence.
The home side had a lucky escape minutes later when Roberts tipped an audacious Ross Montague lob over the bar.
Dagenham hit back with a sucker punch in the 47th minute when Strevens thundered in his third goal in two games, after pouncing on a loose ball on the edge of the Bees' box.
The Daggers should have made it two in the 61st minute when Strevens played Sam Sloma into space, only for the winger to slice his cross over the bar.
Sloma missed the target again in the 65th minute when he fired wide from 15 yards after latching on to a clever Southam through ball.
Poole grabbed the lead for the visitors in the 69th minute when he met a Montague cross and slotted past Roberts from six yards.
The striker should have had a hat-trick a minute later, but shot too close to the keeper with the goal at his mercy.
The Daggers upped the pressure in search of an equaliser, but lacked the finishing skills to punish Brentford late on.