Johnson's effort in the 113th minute left Stockport keeper Lee Jones with no chance, as the ball sailed into the top left-hand corner of the net.
Earlier Stockport deservedly took the lead after eight minutes following some good early pressure.
John Hardiker delivered a teasing ball from the left wing, which was met perfectly by promising striker Jon Daly. He headed home from eight yards to leave Gillingham goalkeeper, Vince Bartram, no chance.
But the visitors did not spend much time licking their wounds and Guy Ipoua levelled matters with a header just 17 minutes later.
Ipoua, who found a yard of space behind County's back line, leapt well to meet a Danny Spiller cross from the right to put the Gills right back in the contest.
The heavens opened in the second half and County toiled to break down their determined opponents.
However, their best chance to take the lead fell immediately after the interval.
But Gillingham captain Paul Smith was in the right place at the right time to head away a Dave Challinor header, after the Stockport skipper had lost his marker at the back post.
Stockport keeper Jones produced a brilliant save to deny David Perpetuini after 72 minutes as the visitors pushed for the victory.
Jones, at full stretch, was equal to a rasping left-footed drive that had looked set to put Gillingham in front.
Rod Wallace should have won the game for the Gills with eight minutes remaining, but the former Bolton Wanderers player fired his shot wide of the post from close range. But luckily for Wallace, it was to be the Gills' night.