Jon Nurse put the Daggers in front with a powerful finish and Danny Green doubled their advantage with a shot into the top corner.
The faltering Addicks grabbed an injury-time consolation effort through Frank Nouble.
Still said: "I never forget who we are, we're Dagenham & Redbridge and to beat Charlton is the greatest result in this club`s history.
"It's the best result we've ever had and to be part of that is a privilege.
"I'm really proud of the players, really proud of the club and really proud that what we've shown our home supporters that we can match the so-called bigger club."
Source: PA
Source: PA