Fallon: Celebrates with Chadwick
Fallon scored either side of half-time to put the Pilgrims ahead before Lee Trundle gave City hope from the penalty spot.
The win moves Plymouth into the Championship play-off places, while City slip to second after Stoke claimed a point against Watford.
The Robins were without Jamie McCombe and Lee Johnson through injury, while Darren Byfield and Liam Fontaine were handed recalls.
Robert Douglas made his debut in goal for the Pilgrims after joining on an emergency loan deal from Leicester on Friday to cover for the suspended Luke McCormick.
City started in a buoyant mood and should have gone ahead after just six minutes but Dele Adebola's header from 12 yards flew narrowly over the crossbar.
Plymouth responded well, though, and threatened shortly after as Peter Halmosi's long-range drive skipped past Adriano Basso's left-hand post.
Halmosi was causing no end of problems for the City rearguard and he tested Basso with a stinging drive from 30 yards after 12 minutes.
Plymouth dominated the opening stages and went close again through Fallon but his header arched just past the right-hand post.
City captain Louis Carey was making his 500th appearance for the club and was having a torrid time against the muscular presence of Fallon as the Pilgrims continued to pose a threat as the half wore on.
Byfield saw his header well kept out by Douglas after 32 minutes as City started to show more attacking threat, while Adebola saw his shot fly horribly wide after latching onto Bradley Orr's neat pass.
City were made to pay for their missed chances in first-half stoppage time as Plymouth took a deserved lead.
Gary Teale created enough space on the right wing to whip in a teasing cross and Fallon was on hand to drill past Basso from 12 yards to take advantage of the Pilgrims' first-half dominance.
City brought on Northern Ireland international Ivan Sproule for the ineffective Brian Wilson at half-time.
City started the half with renewed optimism and Adebola should have pulled them level but he could only guide his header wide from Michael McIndoe's inviting cross.
But any hopes City had of finding their way back into the game were swiftly dashed after 57 minutes as Fallon netted his second.
Teale was again the provider as his corner caused havoc in the penalty area and Fallon pounced from close range to send the visiting support into raptures.
Jermaine Easter saw his long-range effort fly past the post as Plymouth threatened to run away with it soon after.
City were reduced to long-range efforts as they tried in vain to find a way back into the game, with Byfield posing the greatest threat.
But they were handed a lifeline after 72 minutes as Easter inexplicably handled in the penalty area and substitute Trundle slotted home from the spot to reduce the arrears.
Adebola twice spurned clear opportunities to draw his side level soon after with the alert Douglas denying him on both occasions.
They were nearly caught out at the other end, though, as Lilian Nalis ghosted in at the back post only to see his header repelled by Basso.
City piled on the pressure in the dying stages but Plymouth held firm to record a memorable win.