Both sides struggled to create chances in a disappointing first half and Charlton had to wait until the 55th minute to break the deadlock when Scott Wagstaff capitalised on a poor backpass before placing a low shot beyond the reach of Plymouth goalkeeper David Button.
Charlton made the game safe in the 90th minute when on-loan Liverpool striker Nathan Eccleston ended a run into the box with a low shot past Button.
Powell said: "I'm absolutely thrilled. It was a great end to an eye-opener of a week, but I cannot fault anyone connected with this club after what has happened this week, especially the players, my support staff and without doubt the fans.
"The players have done themselves a bit of justice because they haven't exactly been the best at The Valley.
"I've told them in the second half of the season we're going to have to make this place a place where we are going to earn the majority of our points."