The right-back popped up in stoppage time to poke the ball over the line and level things up after visiting goalkeeper Andy Oakes failed to hold a fierce Sam Wood free-kick.
It was the least the home side deserved as they dominated the match and were very unfortunate to be behind.
The visitors looked to have snatched a win with a Richie Foran goal on the hour-mark but reckoned without Osborne's late intervention.
Loan striker Foran nodded a Franz Burgmeier cross past Ben Hamer and could have doubled the lead minutes later when he had only the Brentford goalkeeper to beat but his low shot was pushed out.
Apart from that, Darlington were almost shot-less and Brentford spent large parts of the game camped inside the visitors half.
Osbourne had Brentford's best chance of the first half when he headed a Glenn Poole cross narrowly wide and Mark Phillips also went close for the home side.
Dean Bowditch had an effort fly wide early on and then blazed an effort over the bar, while Rob Purdie had a shot blocked at the other end in one of Darlington's rare ventures forward.
Bowditch also went close early in the second half with a low shot which Oakes saved with his legs and Adam Newton had a fierce shot turned away.
It looked as if Foran's goal was going to give the visitors a win that would have been akin to daylight robbery, particularly when substitute Nathan Elder turned a Ryan Dickson cross wide from close range and Wood missed the target from the edge of the box.
But Osborne nipped in at the death to score a vital equaliser and keep Brentford within touching distance towards the top of the table.