Dickov - in his first start since arriving on-loan from Leicester - almost made it two for the Rams when he shaved a post with a crisp 10-yard drive just before the break.
Owls should have levelled on 55 minutes when a horrible blunder by Dean Moxey allowed James O'Connor to feed Leon Clarke, who with only the keeper to beat, blasted over from eight yards.
Wednesday went close again when a long clearance from Lee Grant was headed onto his own bar by Derby defender Shaun Barker.
Derby went two up on 65 minutes when Croft found Jay McEveley, who let the ball run before drooling a shot beneath the goalkeeper from 14 yards.
And the Rams wrapped up the points with a third goal in time added on when Lewis Buxton upended Rob Hulse inside the box, with the Derby striker dusting himself down before guiding home the penalty low to the keeper's right.
Afterwards Derby coach Gary Crosby said the Rams had been desperate to make amends for their heavy defeat at Cardiff.
He said: "We wanted a good response, and we got one. We have an honest bunch of players.
"It takes a hell of a lot of courage to go out in front of 30,000 people and keep on running and trying, and that's what our players did."
Crosby was delighted with Dickov's performance, adding: "We are absolutely delighted for Paul.
"He's built a good career on hard work and dedication, and that's what he was all about today."
Derby's next game is at east Midlands rivals Leicester in two weeks' time.
Crosby added: "It helps that we are going into a two-week break on a high. It's come at the right time because it gives us a chance to get a few of the injured players fit."
Wednesday manager Brian Laws was livid after his side had failed to perform. He revealed he had warned his players to brace themselves for a Derby backlash after their humiliation at Cardiff midweek.
Law said: "I was concerned because it is a fact that most teams who have had a thrashing generally respond in the right manner but I'm more concerned about how we responded to that threat because I felt we were second best all over the park.
"You can't perform like that and give a team a lift when they are rock bottom, but I felt any nerves were in our team, and not Derby.
"I don't mind losing games if the opposition carve out a wonderful goal, but we gave them their goals on a plate.
"We didn't create anything and we didn't pass the ball. It was our worst performance for quite a long time and there is no defence for a performance like that.
"It was a shoddy performance, it was unacceptable - and I'm not having it. There were no plus points out there for us today.
"We've got a two-week break now because of internationals, and that means the players have got a very angry manager for two weeks."