Ben Hudell was sent off for the hosts after conceding a penalty three minutes before half time, from which Sinclair scored after Elliot Buchanan's effort was parried.
This was an eagerly-anticipated match, but the opening twenty-five minutes failed to live up to expectations with the ball spending much of the time in the air and neither side building a decent spell of possession.
The first real talking point came as Burnham’s Aaron Couch put in a poor tackle on Slough skipper Paul Edgeworth, the Blues player catching him above the ankle. Couch escaped punishment, and minutes later burst free on the break and forced Ricky Perks into a fine full-length save with an effort from 25 yards.
Burnham top-scorer Fennemore fired well wide from distance and, as the game finally began to open up, Slough’s best moment came as the impressive Adam Martin burst forward and fed Sinclair, who got into the box but crossed too deep for either Sonner or Simon Martin to reach.
The resulting corner caused some nervous moments in the Burnham defence, but Edgeworth fired a volley just over the bar.
The opening goal, scored three minutes before the interval, was also drenched in controversy. Simon Martin laid the ball off to Buchanan in the six-yard box, and he was felled by a mistimed tackle from Ben Hudell as he was shaping to shoot. Referee Sam Lewis pointed to the spot immediately, before sending Hudell off, for what looked like a very harsh decision, with the tackle being no worse than Couch’s earlier in the match.
Buchanan dusted himself down to take the penalty, but his effort was easily parried by Paul McCarthy. However Steve Sinclair followed up, tucking the ball underneath the keeper to give Slough the half time advantage in front of a season's best crowd of 507.
With the man advantage, Slough pressed forward in the second half and had much of the play, but were frustrated time and time again by Burnham, who denied them space and the Rebels failed to test McCarthy again.
Slough were then made to pay for not making their advantage count as Burnham won a free-kick on the edge of the box. Paul Coyne and Stuart Swift stood over the ball, but it was Swift who hammered the ball left-footed into the bottom corner, giving Perks no chance.
Sean Sonner headed across goal from a corner, but at the other end, Yiadom Yeboah had to be at his best to deny Aaron Couch. Sonner put another effort wide with a first-time effort as he was played in, and as the game drew to a close, shots from Seeby and Buchanan were well blocked as the Blues threw bodies in the way of the ball.
At the death, Burnham had a golden opportunity to take all three points as Adam Martin lost possession and Fennemore ran clean through – but he failed to get a shot in or square the ball across goal, and Danny Murphy got back superbly to make a challenge before embarking on one last attack.
Burnham Line up
McCarthy, Tilbury, Hudell, Brett, Dunlop, Sear, Coyne, Swift, Fennemore, Bird, Couch. Subs (all unused): Charlery, Dowdican, Hatcher, Hicks, Morris.