Seventeen-year-old Sinclair, on loan from Chelsea, leapt high to convert a cross from David Norris six minutes from time and send a packed Home Park crowd into raptures.
The Pilgrims had led since the 13th minute of the game, when another loan signing, QPR striker Kevin Gallen, won and converted a penalty.
Sinclair, who was nurtured as a schoolboy at Bristol Rovers by Argyle manager Ian Holloway, has already set the FA Cup alight this season with a wonder goal in Argyle's 2-0 fourth-round win at Barnet and is on his way to making this year's competition his own.
Gallen, though, was at then centre of most of what went on at Home Park in a typical cup tie, with bags of atmosphere, passion and wholehearted commitment.
Referee Mike Dean decided that Leacock had tripped Gallen but declined to send the defender off and Gallen netted his first FA Cup goal for nine years.
Derby did their best to unnerve Gallen and Bywater nearly reached the effort low to his left, but the strike had just enough pace on it to carry the ball into the net.
Bywater kept Derby in the game with a tremendous save from Argyle's teenage right-winger Dan Gosling, who curled a left-footed shot after cutting in from the right that the County keeper tipped around the post.
Gallen had a chance to extend Argyle's lead over the Championship leaders midway through the first half when Moore was adjudged to have fouled Hungarian centre-back Krisztian Timar, following a Gallen corner.
Similar protests from Derby followed and Bywater out-psyched Gallen by guessing that the Argyle man would change sides from his first kick and, although there was little wrong with Gallen's kick, Bywater made a magnificent stop.
Timar cleared Stephen Pearson's drive off the line towards the end of the first half, before Derby manager Billy Davies went on the attack at half-time by introducing Giles Barnes and Jon Macken from the substitutes' bench.
Argyle, though, remained the dominant force and Sinclair forced Bywater to tip a dangerous cross over the bar and, from the resulting corner, Mo Camara cleared Timar's shot off the goal-line.
Macken looked the man most likely to bring Derby level and it took a fantastic save from McCormick to keep the Pilgrims ahead.
With the game getting more and more competitive, Moore was booked for a late tackle that floored Argyle captain Lilian Nalis and was then sent off after receiving a second yellow card following a clash with Gallen.
Derby nearly drew level when Steve Howard picked out Macken, but Paul Connolly got back to deny him with a magnificent last-ditch tackle.
As Derby pressed, they left massive spaces behind their defence. Gallen unleashed a rising drive that went only narrowly over, before any late tension for the home crowd was dissipated by Sinclair.
Norris chased a lost cause before crossing perfectly for Sinclair to rise up and send a perfect downward header past the impressive Bywater and in off the post.