Dickinson's near-post effort was enough for Stockport to finally grab a much-deserved victory.
On a glorious afternoon, the home side paraded five summer signings, with Dagenham just showing the one off.
Keeper Conrad Logan, defenders Paul Tierney and James Smith, midfielder Gary Dicker and striker Matty McNeil made their Stockport bows, while front-man Jon Nurse did likewise for the Daggers.
County were the livelier in the opening exchanges, and before the 10 minute mark was up Adam Griffin was denied by a superb Tony Roberts save, and David Poole smashed a smart volley inches wide.
Scott Griffiths shot over for Dagenham soon after and, then just before the half-hour was up, Nurse wasted a great chance when he planted a free header well over the bar from eight yards.
The home side were on top though, and at the other end McNeil should have done much better when he fired a shot straight at the grateful Roberts from 15 yards.
Stockport completed a dominant opening 45 minutes with chances missed by David Poole and Anthony Elding, but Dagenham hung on and the teams went in scoreless at the break.
Stockport created the first chance of the second half when McNeil inadvertently blocked a powerful goal-bound drive from team-mate Jason Taylor.
The home side saw a Griffin effort chalked off for offside on 54 minutes, before Roberts once again came to Dagenham's rescue with another top save from Elding.
Stockport eventually scored the all important and late goal, with Dickinson doing the business for County.