Woody completes late goal blitz from Blues
The Bees led from 20 minutes when Alfie Mawson opened the scoring before Alan Lee started the goal frenzy with an equaliser on 75 minutes.
George Pilbeam was sent-off for the home side after picking up two bookings and Blues made Brentford pay.
Jordan Adekunle put Town in front nine minutes from time, Academy scholar Matt Clarke extended the lead soon after before Woody made it four on 90 minutes to record the U21s' first win this season.
Town included Stephen Hunt, Scott Loach and Freddie Veseli in the starting line-up.