The Pilgrims looked on course for a straightforward three points in the Championship on Tuesday night when Chris Clark netted his first league goal for the club before Bradley Wright-Phillips doubled the advantage before the break.
Nicky Maynard grabbed his first goals of 2010 to level matters, but Argyle had the final say as Fletcher fired home a right-footed shot from 20 yards in the final minute.
The result moves Argyle, who last won a home match against the Robins six years ago, to within four points of third-bottom Scunthorpe, who travel to leaders Newcastle on Wednesday.
The first chance of the game fell to Maynard after just seven minutes, but the striker, who had scored just one goal in his previous 20 appearances, shot wide.
His last goal had come in the 2-1 defeat at QPR in December, a strike that recently saw him pick up the Mitre Goal of the Year award, but he ended his barren run with two second-half goals.
At the other end, Wright-Phillips saw his effort blocked by Louis Carey, who was making his 600th club appearance for the Robins.
On-loan goalkeeper David Stockdale then came to Argyle's rescue to deny Cole Skuse with an excellent parry. Debutant Stefan Maierhofer, who this week joined on loan from Wolves, was on hand to tap home the rebound but his effort was ruled out for offside.
But it was Argyle, who had the worst home record in the Championship prior to kick-off, who opened the scoring with a rare goal from Clark shortly after the half-hour mark.
Clark latched onto a cross from Chris Barker before firing home a sweetly-struck volley at the back post for his first goal of the season.
The Pilgrims could have doubled their lead six minutes before the break but Jamie Mackie's effort was parried by Dean Gerken, who comfortably gathered the ball at the second attempt.
Paul Mariner's men increased their advantage on the stroke of half-time as City failed to deal with a long throw-in from Kari Arnason, leaving Wright-Phillips to score his first goal for the club with a spectacular overhead kick.
City got themselves back into the game within 13 minutes of the restart as Maynard scored from a quickly worked free-kick, and the Robins would have been level soon after had Stockdale not parried away Jamie McAllister's effort.
The Robins finally drew level when Maynard sent a right-footed shot beyond Stockdale 12 minutes from time and the striker almost completed his hat-trick soon after, only to see his header sail just wide.
Jamal Campbell-Ryce was dismissed for a second bookable offence late on and Argyle took full advantage of the extra man with Fletcher's winner.