Constable had only been on the pitch for 11 minutes when he took advantage of some slapdash defending to follow home an equaliser with 12 minutes left.
He was allowed to run into the box unopposed and his shot took a couple of deflections before sneaking past home keeper Ben Hamer.
It was no less than Shrewsbury deserved after matching Brentford a team who had beaten them on their own patch just four weeks earlier.
Nathan Elder should have opened the scoring for Brentford at 21 minutes when he somehow put a shot from point blank range from Ricky Shakes' cross over.
Alan Connell saw a couple of efforts from outside the box saved for the Bees and, in between, Kevin McIntyre reminded Brentford of the danger down the other end when he flashed a shot across the face of goal.
Ryan Dickson was just wide with a curling free-kick as half-time approached while after the break, Dave Hibbert screwed just wide for the Shrews after a good flowing move.
Visiting keeper, Glyn Garner made a great save from Connell on 52 minutes after he side-stepped Colin Murdock's challenge, but Connell got the better of Garner five minutes later by beating him to score off the far post after Glenn Poole's cross broke to him in the box.
Connell should have made it two a minute later but Garner saved well before Shrewsbury started taking the game to Brentford with Ben Davies' belter deflected wide and Hamer saving well from Asa Hall.
Dickson shot over after beating three players and Elder saw his shot blocked by Kelvin Langmead as the game swung Brentford's way again, but on came Constable to equalise for the Shrews.
However, the visitors had to survive two late scares to clinch the point.
First Matt Heywood's 87th minute header was heading for the top corner until it was punched off the line by Garner, and just a minute later Marc Tierney scrambled Elder's header off the line when a goal seemed certain.