United found themselves with a two goal advantage twice in the match thanks to a hat-trick from Shania Hayles.
But Becky Langley saw her side go down to ten players in the second half when leading 3-1 when Amber-Keegan Stobbs was sent off for two bookable offences - something the Addicks pounced upon.
A second-half double from Bradley, accompanied by a Kayleigh Barton goal, was enough for the visitors to pick up a last-gasp point in the Barclays Women's Championship.
The game's first shot on target produced the opening goal of the match through Hayles. The Jamaica international flicked on a long ball setting Amy Andrews free down the right.
United's number 17 drove into the box and unselfishly cut back it back to Hayles, who finished off the move by calmly slotting home into the bottom corner.
Newcastle were on fire in the first half with Stobbs picking up the ball in midfield and after one touch she slipped through Hayles.
The striker raced through on goal before lifting the ball over Jess Gray's leg and into the bottom corner just nine minutes after her opening strike.
Hayles almost had her hat-trick from just three shots on target one minute later as she was picked out by a pinpoint Andrews through ball, but was denied by Gray's outstretched leg.
Charlton looked to have found a route back into the game on the half-hour mark when Kayleigh Barton poked the ball beyond the onrushing Claudia Moan, only to be frustrated by an assistant referee's flag.
However the Addicks finally got themselves back into the game just four minutes after the restart when Bradley fired home after Newcastle failed to clear the ball from their own box.
Despite waves of Charlton pressure Newcastle were handed a chance to restore their two goal lead when Lois Joel was hacked down in the box.
With just over 20 minutes left to play, Hayles stepped up as Kingston Park fell silent and sent Gray the wrong way to secure her hat-trick.
It was almost a fourth moment later when Gray came rushing out of her area to block Hayles' effort and managed to scramble back to her line and save Lumsden's effort from the rebound.
United's task of seeing out the game was made slightly more difficult in the final six minutes of regular time when captain Stobbs was shown her second yellow card for delaying the restart.
And Charlton made their numerical advantage count in added time with goals from Barton and Bradley coming within seconds of each other to secure a point for the visitors.
Newcastle United Women: Claudia Moan, Isabella Sibley, Hannah Greenwood, Charlotte Potts, Demi Stokes, Elysia Boddy, Amber-Keegan Stobbs, Lois Joel (Deanna Cooper 91), Amy Andrews (Katie Barker 65), Shania Hayles (Sophie Haywood 79), Beth Lumsden (Kacie Elson 79)
Subs not used: Grace Donnelly, Hannah Hawkins, Liv Watt, Georgia Gibson, Jasmine McQuade
