Bees goalkeeper Lee saved three spot-kicks in succession to leave Bees striker Gary Alexander with the task of sending Andy Scott's side through to a money-spinning final against Exeter.
It was no more than the Bees deserved after dominating the second half but failing to find the killer touch to end Charlton's Wembley dream.
Brentford centre-back Karleigh Osbourne came closest to breaking the deadlock in regulation time when he slammed a volley against the bar from a free-kick with just five minutes remaining.
Bees' leading scorer Charlie MacDonald then squandered a gilt-edged chance to seal the tie in injury time when he sliced his effort wide of the upright after good work from Alexander. Substitute Sam Wood also flashed a volley wide of the post with almost the last kick of the game.
The closest the visitors came to ending the stalemate was when Lee Martin headed over Charlton danger man Kyel Reid's left-wing cross from close range. Earlier, Reid himself came close with a low drive which was tipped wide by the hero of the hour Lee, who followed up his penalty heroics in earlier rounds with the hat-trick of saves which settled the encounter.