Accrington created the first opening of the game inside the first minute when Shaun Whalley flashed a curling shot wide.
When play switched to the other end Jason Taylor similarly curled a shot over from outside the box.
On 15 minutes Stanley's Andrew Todd did superbly down the right and curled in a great cross, but Paul Mullin could only thump his header wide from eight yards.
Stockport front-man Dickinson was becoming a handful for the Stanley defence, and he barged clear before seeing a low drive saved by Kenny Arthur.
Stockport went in front on 33 minutes after a splendid set-piece effort.
New loan signing Shaleum Logan floated in a lovely free-kick from the left, and there was Dickinson who headed home his 12th goal of the season from close in.
Just before the interval Michael Rose crossed from the Stockport left, but teenager Tommy Rowe volleyed well wide from just inside the box.
Straight after the restart Dickinson should have done better when he disappointingly headed a Logan corner wide, before Stockport missed a penalty on 53 minutes.
Adam Proudlock was felled by Sean Webb, but Rose saw his spot-kick well saved by Arthur down low to the keeper's right.
The visitors did double their lead on 63 minutes though. Rose played a ball into Dickinson, and he simply shrugged off Aswad Thomas before firing home from just outside the Stanley box.
Thomas redeemed himself slightly by clearing Proudlock's looping shot off the Stanley line, with Stockport deserving winners in the end.