Simon Church headed Charlton's second-half winner in a rearranged fifth-round tie at Hillsborough after Wednesday striker Leon Best had cancelled out Callum Harriott's first-half opener for the Londoners.
Charlton, in the bottom three of the Championship, set up a last-eight clash with Sheffield United just two days after giving their survival hopes a boost by beating QPR.
"Everyone was talking about the potential Sheffield derby and we sort of went under the radar," said a delighted Powell, who celebrated in front of Charlton's fans after the final whistle by swinging on the crossbar.
"I know I shouldn't have done that," smiled the former Addicks defender.
"It's been a tremendous 72 hours for us. It was a big, big result against QPR and then to travel up here with some tired bodies."
Wednesday substitute Atdhe Nuhiu's second-half header came back off the inside of the post and momentarily appeared to cross the goal-line before being caught by Charlton goalkeeper Ben Hamer, who also pulled off a fine save to deny Chris Maguire's late goal-bound shot.
"We got the second goal and it was heroic defending time by all the players, especially the keeper who made two or three outstanding saves," Powell added.
"And now we find ourselves in the quarter-finals. We've got a real chance now of getting to the semi-finals, but it will be very tough against Sheffield United, who've had some really good results in the cup.
"It's going to be another tough game, if not harder because people will be looking at us the favourites.
"Nigel (Clough) and his team will be thinking the same thing. They want to get to Wembley. A Football League side will be in the last four and we'll be trying our hardest to make it us."
Wednesday head coach Stuart Gray has never made it to the last eight of the FA Cup as a player or manager and could not hide his disappointment as his side passed up the chance of a memorable clash with their city rivals.
"You can't play like we did in the first half and expect to win games of football," he said.
"It was 27 minutes I think before we got a cross into the box, about 40 minutes before our first shot, 41 minutes before our first tackle and 44 minutes before our first corner.
"You're not going to win games of football playing like that. Whether it was nerves that took over, that's the worst 45 minutes of football I've been involved with at the club.
"They've missed a great opportunity to look back over their careers and have on their CV a quarter-final of the FA Cup against your local rivals.
"I just couldn't see it coming. We had a go second half and built up a bit of momentum, but I've said to the players they have to start from the word go.
"It was an FA Cup quarter-final at stake and it's a big disappointment."