He pounced to ram home after Stephen Gleeson's thumping shot had rebounded away off the crossbar.
Stockport had the better of the opening half, but failed to turn their territorial supremacy into that all-important goal.
Loan midfielder Gleeson fired over disappointingly from 12 yards, but that was followed by Notts' best chance of the opening period on 20 minutes.
Junior Mendes fired in an angled cross-cum-shot that just missed David Pipe's right-footed lunge at the far post.
Stockport striker Dickinson then twice went close as he continued his search for a first goal of the season.
Firstly his acrobatic overhead kick just missed the post and then he shot low and just wide following skipper Tony Dinning's neat through ball.
Magpies midfielder Andy Parkinson tried his luck a minute before the break but his arrowing drive was well saved by James Spencer.
The second half began in dream fashion for youngster Anthony Pilkington, who headed Stockport in front clinically from Adam Griffin's tempting cross.
The 18-year-old celebrated his first senior goal in just his second appearance following his recent move from Atherton Collieries.
Notts then sprang into life and on the hour-mark Mendes was unfortunate when his close-range shot hit the base of a Stockport post.
Minutes later Jason Lee saw a 25-yard effort well gathered by Spencer.