Sinclair's 56th-minute strike took a wicked deflection off Clint Dempsey on its way into the back of the net.
And it was Dempsey who left Wales wishing he had never crossed the Severn Bridge after his spot-kick was saved by Swans keeper Michel Vorm.
But the hosts fully deserved to have fortune on their side before substitute Graham made it 2-0 in the 90th minute to secure three precious points and move the hosts towards to mid-table security.
Swansea boss Brendan Rodgers named Steven Caulker, Luke Moore and Wayne Routledge in the starting line-up to give his goal-shy team a boost.
And it was Moore and Routledge who looked most threatening during a first half the home side dominated without threatening Mark Schwarzer's goal.
Sinclair was just unable to get a shot away after linking with Leroy Lita and headed wide from a Routledge cross in a bright opening.
The Cottagers enjoyed a rare foray as Ashley Richards denied Andrew Johnson after the striker got the better of Caulker.
But the action quickly swung back down the other end other end, with Moore seeing a swerving strike from just outside the box clip the woodwork.
Fulham then survived two penalty appeals against John Arne Riise.
The Norway international was awarded a fortunate free-kick after appearing to haul down Lita, before being adjudged not to have handled a Routledge cross.
Schwarzer made the first real save of the afternoon on the half-hour mark, holding a Sinclair shot, and Dempsey headed wide as Fulham finally carved an opening.
Aussie stopper Schwarzer required treatment early in the second half after suffering a heavy landing when falling over Lita.
He managed to continue and the home crowd's frustrations were beginning to grow before the game burst into life.
Schwarzer's punch only went as far as Routledge and his scuffed effort was turned goalward by Sinclair before ricocheting off the unfortunate Dempsey.
Fulham responded to falling behind but Vorm, who had been almost a spectator up to that point, denied Bryan Ruiz from eight yards and then held an angled drive from Dempsey.
It was suddenly all Fulham as Riise fizzed a free-kick just wide and Vorm raced out to deny Johnson.
But with Fulham searching for an equaliser, a rapid Swansea break almost saw them double the lead.
Sinclair teased full-back Stephen Kelly before teeing up substitute Kemy Agustien, whose side-footed effort beat Schwarzer but struck the post.
Kerim Frei then went tumbling over Richards in the box and referee Jon Moss pointed to the spot, only for Vorm to make a stunning stop to deny Dempsey.
And there was still time for Graham to grab his fifth goal of the season from Mark Gower's corner in injury-time.
Swansea: Vorm, Richards, Williams, Caulker, Taylor, Routledge (Dyer 89), Britton, Gower, Sinclair, Lita (Graham 76), Moore (Agustien 69). Subs not used: Tremmel, Dobbie, Monk, Bessone. Booked: Vorm.
Goals: Sinclair 56, Graham 90.
Fulham: Schwarzer, Kelly, Hangeland, Senderos, John Arne Riise, Dembele (Frei 77), Davies (Orlando Sa 79), Etuhu,Dempsey, Ruiz, Johnson. Subs not used: Etheridge, Gecov, Bjorn Helge Riise, Hughes, Briggs. Booked: Senderos.