But there was a stormy finish at The Valley as Oldham had Jason Jarrett sent off for a second yellow card three minutes from time.
The defender angrily threw the ball at Charlton boss Phil Parkinson as he trudged off, clearly believing his reaction to a foul on Kyel Reid influenced referee Rob Shoebridge's decision.
Players from both sides then squared up as Jarrett headed to the tunnel, but once calm was restored the Latics held on.
Furman struck on the stroke of half-time, giving Addicks keeper Rob Elliot no chance with his sweet right-footed finish to cancel out Scott Wagstaff's 24th-minute strike.
The opener followed a deep cross from Simon Francis which was knocked down by Pawel Abbott. And as Lee Martin and Jarrett battled for the ball, it rebounded across the box for Wagstaff to pounce.
Charlton did most of the pressing in a pulsating contest, but it was Oldham who had the better chances.
Oumare Tounkara hit the post twice before forcing a superb 70th-minute save from Elliot after being played in by Warren Feeney.