The Gills were heading for defeat until substitute David Graham brought them level in the 85th minute by heading home Mark Bentley's cross.
And Dickson had the home fans in raptures when he fired home the winner from 15 yards in the 89th minute.
The home side had gone ahead in the 19th minute when Aaron Brown's fizzing 20-yard free-kick in the 19th minute pierced a hole in the Rovers wall and went under unsighted goalkeeper Steve Phillips.
But Paul Trollope's men were back on level terms on 27 minutes in almost identical fashion.
Referee Darren Drysdale controversially awarded a free-kick on the edge of the box after Simon King appeared to bundle over Craig Disley inside the area.
This infuriated the visitors but they were soon celebrating when Rickie Lambert fired the free-kick into the roof of the net via a vicious bounce off a defender.
There was no fortune about Rovers' second goal when, just 30 seconds into the second half, David Pipe fired a perfectly executed volley into the far corner after reacting fastest to Richard Walker's flick in the area.
Gillingham proceeded to pepper the opposition area with a succession of crosses and were denied by Aaron Lescott's goalline clearance after Philips had fumbled Nicky Southall's corner.
Their plan was proving fruitless until the introduction of Graham, who sneaked in unnoticed in the area to head home Bentley's perfect cross.
Rovers were rocked and any hopes of salvaging a point from the game diminished with a minute of normal time remaining when Dickson lashed home Delroy Facey's flick.