Second-half goals from Jordan Hugill and Marcus Tavernier made it five defeats from six matches under Thomas Frank for Brentford.
Hugill broke the deadlock just before the hour mark, slotting home Muhamed Besic’s slick pass from close range as the Bees defence were caught napping.
And Tavernier made it two six minutes later, getting ahead of centre-back Chris Mepham to glance Dael Fry’s right-wing cross home just inside the far post.
The setback brought an immediate response from Brentford but Judge’s low 20-yard drive across keeper Darren Randolph was not enough to prevent another Bees loss.
Boro, unbeaten in six, started the brighter and could have gone ahead inside four minutes when Bees keeper Daniel Bentley flapped at a cross which fell for Aden Flint, who blazed over on the half volley from close range.
Boro posed the early threat and, midway through the half, a loose Brentford ball set Hugill free in a one-on-one chase with Mepham, but his rising drive was stopped by Bentley.
Bees winger Sergi Canos should have done better with a free shot just before the half hour, but his effort lacked conviction and bobbled wide of the far post.
Moses Odubajo came closest for the hosts minutes later, his dipping drive from distance after a neat one-two on the edge of the box with Neal Maupay being tipped over by Randolph.
Brentford lacked a real cutting edge throughout and only started to bare their teeth once they were two behind, Lewis Macleod and Canos forcing Randolph into comfortable saves.
Maupay was forced to feed off scraps against Boro’s two big centre-backs, but came close with a powerful header from substitute Emiliano Marcondes’ right-wing cross.
Judge reduced the arrears, catching the visitors napping from a corner, and two minutes later Canos wasted a gilt-edged chance to equalise when he fired tamely at the keeper from close range.
The diminutive Irishman could have levelled with minutes remaining but his speculative far-post volley flew high and wide as Brentford threw everything forward in search of a foothold in the game.
The last word went to former Arsenal youngster Josh Dasilva, who drove forward and unleashed a swerving low drive which Randolph did brilliantly to tip wide.
Brentford, without several key players through suspension and injuries, now look to Tuesday night and the visit of Sheffield United to build on the final 20 minutes.
This post was last modified on 24/11/2018