The Addicks took the lead after a dominant start to the game when Jodie Hutton found the bottom corner from outside the area.
However, from that point it was all United, with the equaliser coming just after the hour mark when Shania Hayles scored her 12th league goal of the season following a foul on Beth Lumsden.
After a period of soaking up some early Charlton pressure, it was United who carved out the first chance after five minutes, but Hayles' close-range header was down the throat of home goalkeeper Whitehouse.
It was the hosts who took the lead after largely bossing the opening 20 minutes. Newcastle failed to clear their lines with the ball eventually falling to Hutton on the edge of the area, and she fizzed a low effort beyond Claudia Moan's outstretched arm.
United responded very impressively two minutes later when Emily Murphy drove across the Charlton box before seeing her effort tipped behind the post by a scrambling Whitehouse.
Newcastle began to turn the screw after going behind and looked to have levelled things up with ten minutes left of the first half. A neat layoff from Hayles offered Lumsden the chance to fire at goal, but a stunning diving save from Whitehouse maintained the hosts' lead.
However, just two minutes before the break, Charlton looked set to double their advantage. A stray pass from Freya Gregory was latched onto by Ellie Brazil, who raced through but was thwarted by Moan.
The Lasses were the side pushing forward and trying to find a way back into the game with a few half-chances going begging, and Charlton were pinned back in their own half.
As the clock ticked over into the 65th minute, United's pressure proved too much for the home side as Charlotte Newsham pulled back Lumsden in the area, handing Hayles the chance to score from the spot.
The Jamaica international was forced to wait a while as Charlton cleared the box, but she made no mistake in converting her spot kick, sending Whitehouse the wrong way and finding the bottom left corner.
It was United who almost added a quickfire second goal to take the lead when Hayles peeled off to the back post and met Rachel Furness' delicate delivery, but she could only head into the arms of Whitehouse from six yards out.
In the space of 60 seconds, Newcastle came within touching distance of scoring on two occasions. Firstly, Lumsden's free kick from 25 yards out looked destined for the top corner, only for Whitehouse to get fingertips to her effort and push it over the bar.
From the resulting corner, Newcastle recycled the ball back into the penalty area with Demi Stokes rising highest - but her looping effort bounced off the post to safety as Charlton struggled to get out of their own final third.
Despite the constant pressure from Newcastle, Charlton almost found an unlikely winner through Brazil, who created space for herself on the edge of the area but could only drag her effort wide.
Once again, Lumsden thought she'd beaten Whitehouse and secured all three points with one minute left when she cut onto her left foot and whipped an effort bound for the roof of the net, only for the Charlton shot stopper to leap into the air and push it behind.
Newcastle United Women: Claudia Moan, Charlotte Wardlaw (Liv Watt 59), Hannah Greenwood, Deanna Cooper, Demi Stokes (c), Rachel Furness (Isabella Sibley 90+3), Jasmine McQuade (Lois Joel 59), Emily Murphy, Beth Lumsden, Shania Hayles (Katie Barker 90+3), Freya Gregory (Amy Andrews 59)
Subs not used: Grace Donnelly, Hannah Hawkins
