Dickinson was only in Jim Gannon's starting line-up because Adam Proudlock, who had been first-choice to start up front, reported in sick.
And he responded with goals in either half to take his tally to five for the season.
Dale were given some hope by Adam Le Fondre's 68th minute penalty against his former club.
County boss Jim Gannon said: "Every point is valuable and in the context of the game it was a big three points.
"There were a lot of things that disappointed me as well though. We gave the ball away far too much which put us under pressure and our defenders had to work far too hard.
"The win came down to a couple of good saves, a couple of good blocks and good solid defending."
Keith Hill's side had been straight at the visitors from the start in a manner that suggested they were anxious to repeat the seven goals they put past County at Edgeley Park earlier in the year.
Le Fondre, who had been recalled for the visit of his hometown club, had a goal ruled out early on.
Tom Kennedy's 16th minute free-kick brought a great save from Conrad Logan, who pushed the ball onto the crossbar.
But once County had survived the early onslaught they took the lead with their first chance of the game. Adam Griffin raced at the Dale defence in the 27th minute and squared the ball for Dickinson who sidefooted the ball past former Stockport keeper James Spencer.
Injury-ravaged County had lost Ashley Williams and Dominic Blizzard, each for at least six weeks, and Gannon had hoped to bring in Michael Raynes in central defence, but the youngster was still suffering from the bug that had kept him out of Saturday's game against Peterborough.
Matty McNeil moved back from the forward line to play alongside Gareth Own and was instrumental in keeping Dale at bay for long periods.
County full-back Paul Tierney was lucky to stay on the pitch when he appeared to headbutt Glenn Murray, but escaped with a yellow card. Dickinson scored again in the 52nd minute when his glanced header guided Tierney's free- kick past Spencer but Murray should have pulled one back in the 61st minute.
But in the 68th minute Jason Taylor stuck out a hand to stop Le Fondre's flick in the County box and the striker stepped up himself to drill home the penalty.
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Fair play to McNeil he was the best player for us on Sat and last night he had a good game at centre half, in our present injury situation I'd keep him there. The game itself was a welcome 3pts we hung in there and rode our luck, 2 well taken goals from Dicko, the second a tremendous free kick from Tierney, the only decent thing he did all night and should have been sent off, likewise Taylor who got booked and now faces suspension. Pleased to see Rosey back in action, his experience showed when we were under pressure towards the end.
Extremely fortunate to win but fair play, 3 points is 3 points. I don't believe there is any danger of us going down and never have done. It's more the awful style of play.
Good to hear Rose played OK, I'm sure Jimbo would rather have brought him back a bit more in stages after his layoff rather than throw him in at the deep end in what is a pretty bad situation injury-wise.
As for McNeil doing a job on defense, who saw that coming? I had no idea he was that versatile, suppose he must have played defence in the past, and it would seem in a significant number of games, going by reports of how he has performed when called upon to fill that role recently.