Rodgers was dismissed from the Madjeski Stadium hotseat last term after less than half a season in charge.
But the ex-Watford boss seems to have been rejuvenated after taking the reins at Swansea in the summer and the south Wales outfit leapfrogged Reading to move into the Championship play-off places courtesy of their third consecutive away win.
Reading enjoyed the better of the early exchanges but the Swans settled into a slick passing rhythm to establish control midway through the first half.
Their pressure was rewarded when Sinclair got the better of Royals goalkeeper Adam Federici to claim his ninth goal of the season.
Nathan Dyer rattled the crossbar for Swansea in the second period before they were forced to withstand a late surge of Reading pressure.
The hosts made a bright start and had an early appeal for a penalty turned down when Shane Long's header from Ian Harte's corner appeared to strike a Swansea arm.
Republic of Ireland striker Long had the game's first clear opening in the 10th minute but glanced Jem Karacan's cross narrowly over from six yards.
Swansea midfielder Mark Gower hobbled out of the action in the 20th minute and the introduction of his replacement David Cotterill appeared to rally the visitors who had struggled in the early running.
Rodgers' side began to play incisively through the midfield and they started dominating possession.
Dyer was in impressive touch for the Swans, but referee Oliver Langford was unimpressed when he took a tumble in the Reading penalty area after 25 minutes.
Speculative efforts from Karacan and Hal Robson-Kanu were deflected behind as Reading attempted to maintain a threat, but it came as little surprise when Swansea took the lead 10 minutes before the interval.
A searching pass from Ashley Williams found Cotterill in the Reading half and his through ball picked out Sinclair.
The former Chelsea forward rounded Federici and slotted home.
Swansea's advantage could have been greater at half-time as Dyer and Stephen Dobbie both dragged wide when well placed.
Brian McDermott's Reading approached the second half with renewed vigour, Robson-Kanu and Jobi McAnuff threatening and probing from wide areas.
But Swansea came agonisingly close to doubling their advantage in the 54th minute.
Dyer exchanged passes with Dobbie to race clear of the Royals backline and his angled shot cannoned back off the crossbar with Federici beaten.
The Australian keeper then saved from Dobbie after he was played through by Andrea Orlandi and the former Queen of the South striker saw another effort hacked off the line following Williams' knock down from a Cotterill corner.
Reading entered the final quarter of the game with a flurry of corners, and Jimmy Kebe should have levelled when a deep delivery evaded everyone and he curled his shot over.
Swansea managed to hang on, but not before substitutes Simon Church and Noel Hunt had headed over from close range.