Brentford are now 11 games unbeaten and dominated throughout as they maintained their place at the top of the table.
It took just nine minutes to open the scoring, with Judge firing home emphatically into the far side of Ben Garratt’s net.
After missing a couple of chances to double his tally, the winger’s second goal was well worth the wait as he whipped a stunning effort into the top corner from 25 yards in the 52nd minute.
Crewe struggled to get a foothold in the game as Brentford’s midfield dominated possession.
It was a central midfielder netted the third, with Adam Forshaw’s 35-yard shot arrowing into the top corner after 62 minutes.
Alex pulled one back in the 70th minute through Chuks Aneke when a penalty was awarded for handball, but it was only a consolation.