Alan Pardew's side threatened to snatch all three points after Matt Thornhill had been sent off in the 84th minute for picking up his second booking.
But, prior to that it had been Colin Calderwood's team who were on top, in an entertaining 0-0 draw.
Forest were the first to pile on the pressure as they won a flurry of four early corners that Charlton desperately defended, with Nicky Weaver having to make a smart stop to deny Robert Earnshaw, after the striker had headed a Wes Morgan flick goalwards from close range.
Some determined closing down from Nathan Tyson forced a mistake from Jonathan Fortune at the back, giving Lee Martin the chance to fire in an angled shot that Weaver palmed away.
The winger then went even closer as he danced through three defenders into the box before bending a shot narrowly wide of the post, while Earnshaw also came close with a clever chip that was not far off target.
Weaver was tested again as Chris Cohen let fly with a powerful, crisply struck effort that the keeper gathered at the second attempt.
Paul Smith still had to make a brave save at the other end, sliding in at the feet of Paul Holland, after the Forest defence had twice failed to clear a loose ball in the box.
It was largely Forest who were on top, with James Perch sending a header narrowly over from a Martin free-kick, although Lloyd Sam also went close and Hameur Bouazza tested Smith with a low drive.
Chances were initially fewer in number after the break, with Martin spurning a decent chance for Forest, firing a weak shot straight at Weaver after being sent into the box from the left by a clever pass from Cohen.
A Cohen corner picked out the tall figure of Wes Morgan as he rose highest on the edge of the six-yard box, but his firm downward header was directed straight into the back of Kelly Youga.
Andy Cole almost had the chance to score with his first touch as Tyson sent a lofted cross-cum-shot to the far post, where the veteran just failed to make contact as the ball dropped narrowly wide.
There was more agony for Forest as Thornhill flashed a low ball right across the face of goal where Tyson and Cole were both inches away from converting at the far post.
The home side were then reduced to ten men after youngster Thornhill slid in recklessly Grant Basey to earn his second booking of the game.
And it took a world class save from Smith - who plunged low to his right to somehow get a hand to a header from Andy Gray that looked certain to find the bottom corner - to deny Charlton a late winner.