But on-loan Scunthorpe winger Winn smashed the ball home three minutes into stoppage time to ensure Saturday's clash at the Gateshead International Stadium ended honours even.
After a dismal first half, the Blue Square Premier side made the early running with Price saving Daryl Clare's point-blank shot. But from the resulting corner, the Bees keeper dropped Winn's set-piece into his own net.
However, Andy Scott's side hauled themselves level when former Newcastle striker Cort headed Ryan Dickson's cross past Paul Farman.
And the visitors got their noses in front when O'Connor crashed home a volley from the edge of the area after Gateshead failed to clear Sam Saunders' free-kick.
Bees defender James Wilson headed against the bar from eight yards before Price's late save from Kris Gate looked to have earned the Londoners victory. Yet Winn had other ideas, lashing the ball into the roof of the net after Brentford failed to clear Armand One's flick-on.