The home side were reduced to 10 men in the opening seconds of Saturday's game when White saw red after an aerial challenge with Damien Spencer which saw the Bees striker carried off.
It took until the 35th minute for the Bees to go ahead when the home defence failed to clear a Sam Wood corner and Alan Bennett fired in, before former Darlington loanee Billy Clarke doubled the lead two minutes before the break when he glanced in a close-range header from a David Hunt free-kick.
But the home side responded with a Franz Burgmeier free-kick that deflected off the Brentford wall and against the crossbar seconds before the interval.
Brentford increased their lead when Clarke fired in a low left-foot drive into the far corner of the net from 15 yards in the 53rd minute.
Darlington pulled a goal back in the 59th minute when Pawel Abbott held off his man before firing in from close range under the body of Brentford keeper Ben Hamer.