Rafael van der Vaart had given the visitors the lead a minute before the break on Saturday as he capitalised on a defensive error from Joe Allen to sweep home with the help of a deflection from Ashley Williams.
Swansea had not managed to pick up a single Premier League point this season after conceding the first goal, but they bucked the trend as a period of heavy pressure was rewarded when Brad Friedel hesitated when dealing with a deflected Angel Rangel cross for Sinclair to nip in and claim his fifth goal of the campaign.
Tottenham had lost just once in the league since the end of August but this result will have felt like a defeat, especially in the wake of Manchester United's shock defeat to Blackburn earlier in the day.
But few would argue their hosts were worthy of taking something from the game, and they will enter 2012 five points clear of the relegation zone after being tipped by many as certainties to go down.
While manager Brendan Rodgers' decision to field Luke Moore and Stephen Dobbie suggested he had one eye on Monday's trip to Aston Villa, there was no sign of that from his players who took the game to their high-flying visitors from the off.
Nathan Dyer almost gave Swansea a dream start inside four minutes after Mark Gower had dispossessed Scott Parker and switched play. The winger got away a curling left-footed effort that Friedel did well to palm away from his top corner.
Dobbie then dallied over picking out a fellow white shirt in the box having come off best in a challenge with Kyle Walker, before Sinclair scuffed his strike after Gower's corner had found him unmarked.
Those near misses roused Spurs from their early slumbers, and Luka Modric fizzed an effort narrowly wide of Michel Vorm's right-hand post after 14 minutes.
Vorm then denied Emmanuel Adebayor after lovely play between Modric, Gareth Bale and Van der Vaart, although the effort would not have counted as the flag went up for offside.
Spurs should have led after 32 minutes after a horrible pass from Dyer played Emmanuel Adebayor in on goal, but Vorm added to his growing reputation with a superbly timed sliding challenge to take the ball away from the Togolese forward.
Bale, the recipient of so much praise this week, had to this point been a peripheral figure, with young Jazz Richards doing a commendable job of limiting the Wales star's forays down the flanks, while the Swansea midfield were doing just enough to crowd his space when he looked inside.
And his disappointing half was summed up when he was booked for diving over Gower's trailing leg just before half-time.
But the visitors took the lead just a minute before the break as Benoit Assou-Ekotto's cross was turned back into danger by Allen and turned home by Van der Vaart.
Tottenham almost doubled their lead 60 seconds after the restart as Younes Kaboul was given far too much room to advance, the centre-half being denied by Vorm after a neat exchange with Adebayor.
Swansea were looking vulnerable to Tottenham's rapid counter-attacks and a misjudged Allen pass allowed Adebayor to tee up Van der Vaart, Vorm being forced to make the block low to his right.
Indeed the visitors for the first time in the contest looked genuinely comfortable, and Bale prodded just wide as Swansea struggled to contain the seamless movement of the likes of Modric, Van der Vaart and the assured presence of Sandro.
But the home side came within centimetres of a leveller in the 66th minute as Rangel's volley from Allen's corner flew just wide of the target.
Spurs skipper Scott Parker limped from the field with an injury, while Swansea introduced top scorer Danny Graham from the bench, and the pattern of the game suddenly changed.
Graham diverted a fierce Sinclair drive over the bar and it took a fantastic clearance from Walker to deny the striker a tap-in as Dobbie stabbed the ball across the box.
Modric blocked a Sinclair effort as Swansea piled forward, and the pressure yielded an equaliser with just six minutes to play.
Rangel's ball into the box deflected off Assou-Ekotto and as Friedel hesitated the ball broke loose for Sinclair to fire home from close range.