Substitute Dickinson headed in David Poole's cross in the 64th minute - his second goal of the season following last week's winner against Dagenham and Redbridge.
But with 12 minutes to go Lester scrambled the ball over the line from Jamie Winter's corner and, although both sides had chances to win it, a draw was a fair result.
Earlier, the visitors were first out of the blocks and good work by Matty McNeil presented Anthony Elding with a chance to open the scoring.
McNeil chested a long ball into the former Boston man's path on the edge of the area - but his first touch was poor and Spireites' keeper Barry Roche was able to block at his feet.
Chesterfield had their moments too during the opening period and on the half hour the lively Lester thought he had given his side the lead when he fired past Conrad Logan but the former Forest striker's celebrations were cut short by the assistant referee's flag.
The second half began with both sides looking to break the deadlock. And it was the Hatters who did so through Dickenson 19 minutes after the restart.
The tall front man beat Aaron Downes to power a header past Roche - but clashed heads with the Australian defender as he did so.
Both players had to go off for treatment and although Downes had to be substituted, Dickinson was able to return with three stitches in a cut above his left eye.
It seemed Dickinson's bravery was going to be enough to secure the win for Stockport but with the minutes ticking down Lester forced home the equaliser after a goalmouth scramble to earn a share of the spoils.