Charlton forward Clarke registered his sixth goal in eight games when he headed a right-wing cross from Christian Ribeiro past Brentford goalkeeper Simon Moore.
Until then Scunthorpe, who finished the game with 10 men after having skipper David Mirfin sent off for a second bookable offence in the closing minutes, had looked to be heading for a fourth successive defeat.
But for Brentford it was a missed opportunity to clinch their first away win of the season. The visitors, now unbeaten in seven, looked to have the game well in control.
Brentford should have broken the deadlock in the 32nd minute when Clayton Donaldson blasted wide from the penalty spot after his own shot had been handled by Mirfin.
The Bees did snatch the lead in the 43rd minute when Harry Forrester danced his way far too easily through the home defence to score with an excellent finish into the bottom corner and it looked a worthy winner until Clarke's late reply.