Chris Clark and Bradley Wright-Phillips had given struggling Argyle a 2-0 half-time lead and City's second-half comeback was thwarted ultimately by Plymouth captain Carl Fletcher, who scored in stoppage time to earn a vital win for the Pilgrims.
City had started the more promising of the two sides, and Maynard and Marvin Elliott both went close with low drives from the edge of the penalty area.
The Pilgrims were given a let-off midway through the first half when home goalkeeper David Stockdale failed to hold Cole Skuse's shot and debutant Stefan Maierhofer tucked the ball home from a couple of yards.
However, the tall Austrian had strayed into an offside position and had been spotted by the referee's assistant.
Argyle made the most of their reprieve and were two goals ahead by the interval. First, midfielder Clark scored his first league goal for the club, more than two years after his debut, volleying home Chris Barker's delicate chip to the far post.
Then Wright-Phillips showed an acute striker's instinct to score his first goal for the Pilgrims on his full home debut, having sat out most of the season with injury, wrong-footing City goalkeeper Dean Gerken with an overhead kick.
There was still time before the break for City to respond with a fierce shot from Maynard, but Stockdale beat the ball out.
It was Maynard who reduced the arrears just before the hour mark, when Clark fouled Maierhofer to concede a free-kick from which a City move left the Argyle defence bamboozleed.
Only a brilliant save by Stockdale from another free-kick, this time taken by Jamie McAllister, kept the Pilgrims ahead.
Stockdale was unable to deny Maynard in the 78th minute, when Maierhofer induced a mistake from Reda Johnson, allowing Maynard to run in and slot home for the equaliser.
A frantic finish saw Jamal Campbell-Ryce dismissed for two yellow cards before Fletcher fired home for the win Argyle so desperately wanted.