Johnson netted, following up his spot-kick after Swans goalkeeper Dorus de Vries had saved the initial attempt.
That came after de Vries had initially saved a spot-kick taken by Jamie Mackie, but had been adjudged to have moved early, necessitating a re-take.
A game between the two lowest-scoring teams in the Championship was hardly destined to produce a goalfest, and so it proved.
City manager Paulo Sousa made five changes to the side that had drawn 1-1 against leaders Newcastle on Saturday, and sacrificed some purist principles to take into account Home Park's divot-strewn surface.
Argyle's on-loan goalkeeper David Stockdale twice came to the Pilgrims' rescue in the opening 15 minutes, both times thwarting Gorka Pintado with smart stops after the Spaniard had broken the last line of the defence.
Argyle's sole shot on target in the opening 45 minutes was a limp long-range effort from Northern Ireland international midfielder Johnson.
Otherwise, there was little for the home crowd to get excited about, and the biggest cheer of the opening half came when unfortunate referee Keith Hill was smacked in the face by the ball and had to call a halt to the game while he received treatment.
The second half was 40 seconds old when Swansea took the lead, Angel Rangel's quick raking forward pass split the Argyle back four and Darren Pratley latched on to it and banged the ball past Stockdale with aplomb.
Pratley nearly repeated the dose shortly afterwards following an identical move, but this time Stockdale saved well.
As so many others have done this season, Argyle found the Swans' defence hard to break down, and rarely threatened an equaliser despite plenty of late pressure, until Andrea Orlandi handled Yannick Bolasie's corner to concede the penalty from which Johnson eventually scored.