It took Argyle just 20 minutes to take the lead, when Stonebridge headed home a set piece from 6 yards out.
Plymouth nearly broke 7 minutes later, but Justin Whittle halted their attacker in his tracks on the half-way line, and the City skipper's name went into referee Clive Wilkes' book.
In the 61st minute Richard Sneekes came off the bench in place of Mark Greaves in midfield but failed to make much of an impact.
Paul Musselwhite had to pull off a great one-handed save, before City finally looked to get into the game with less than 20 minutes to go.
David Beresford and Ryan Williams were brought on and Brian Little changed to an attacking 4-4-2 formation.
This caused the home side problems as the Tigers started to take the game to them, and Lawrie Dudfield came close with our only shot on target five minutes from time.
However, the Pilgrims withheld our late pressure and took all three points in front of a crowd of 9134.
City drop further down the table, and are now joint 6th/7th with Kidderminster on 41 points. Our better goal difference gives us the higher of the two spots, two points behind Shrewsbury and Rochdale in joint 4th/5th. Mansfield are a point ahead with 44, and are in the last of the automatic promotion places. Luton are 9 points ahead of the Stags but have played two games more than them. Plymouth are still top with 55 points.
City line-up (5-3-2): Musselwhite; Petty, Whittle, Goodison, Wicks, Holt (Beresford); Greaves (Sneekes), Whitmore, Johnsson (Williams); Alexander, Dudfield.
Un-used subs: Glennon, Mohan.