After coming on as a substitute on the hour mark, teenage striker Sinclair, on loan from Chelsea, triggered a decisive spell from the Championship side to enable the visitors to emerge triumphant.
Prior to the arrival of Sinclair, Plymouth were largely second best to the hungry Bees and the League Two side were dominant for large periods of the game - without properly testing Argyle keeper Luke McCormick.
In front of a buzzing home crowd, Barnet began their first-ever FA Cup fourth round tie with a real sense of urgency and belief that translated into a sizzling match played at breakneck speed.
Led by the central midfield combination of feisty captain Nicky Bailey and 41-year-old Andy Hessenthaler, the League Two side rattled their Championship opponents with some full-blooded challenges and appeared to be determined to make the most of the occasion.
Plymouth, on the other hand, looked out-of-sorts and struggled to adapt to the frenetic tempo of the game with Barnet showing a fearless streak every time the home side crossed the halfway line.
The front duo of Jason Puncheon and Adam Birchall linked up well and their mixture of pace and sheer hard graft put Plymouth on the back-foot throughout the first half.
Kicking down the slope, Barnet's slick exchanges led to a series of crosses being pumped into Plymouth's area, but McCormick kept the scoreline goalless with some competent handling.
In key areas, Barnet's inexperience proved costly when Puncheon's indecisiveness meant his cross was gathered up by McCormick when it should have led to an opening goal.
Barnet kept plugging away and, just before half-time, had another gilt-edged chance to seize control of the game with a goal when defender Joe Devera delivered an inviting cross from the right flank that found Birchall.
Even though his marker towered over him, Birchall managed to send his header goalwards, but with insufficient power to really trouble McCormick.
After the break, Barnet continued to dominate the tie with a passionate brand of play, but they could not translate possession into goals against an off-key Plymouth side.
Barnet boss Paul Fairclough was incensed when Plymouth right-back Paul Connolly appeared to aim an elbow in the direction of Puncheon, but referee Walton let play continue.
The incident triggered off some crude challenges around the halfway line and prompted a lapse in concentration from the home side that led to Plymouth snatching the lead.
Against the run of play, Pilgrims midfielder David Norris swept in from the right flank into the congested area and was fouled by Devera with Walton pointing to the spot.
Hasney Aljofree kept his nerve and clinically sent his penalty past Ross Flitney - even though the Barnet keeper had guessed the right way.
Then, with seven minutes of the contest remaining, Sinclair decided to take matters into his own hands and, after picking up the ball deep inside his own half, embarked on a thrilling run up the pitch.
After hurdling past two desperate Barnet challenges, he closed in on goal and swept a neat finish past Flitney to kill off any lingering hopes of a Bees' equaliser.