Bostock, signed on the eve of the campaign on a loan from Tottenham, laid the foundations for victory by rattling home a stunning 30-yard strike to help Nigel Pearson mark his first game in charge with a win.
Returning captain Ian Ashbee made the points safe in the 49th minute, punishing Dorus De Vries' error, as new Swans boss Brendan Rodgers began his new job with a defeat.
Swansea dominated the opening stages, with David Cotterill rattling a free-kick against the bar, but Bostock helped his team shake off any relegation induced hangover following demotion from the Premier League.
The Tigers were without Craig Fagan and James Harper, while the saga surrounding the jettisoned Jimmy Bullard saw the midfielder left out of the squad altogether.
Nolberto Solano was handed his Tigers debut at right-back and Ashbee made his first start since recovering from a career-threatening knee injury last year.
Swansea were dealt an immediate blow when first-choice right-back Angel Rangel was forced out of the starting line-up, so Albert Serran came in as a last-minute replacement.
But they started brightly and had sight of Matt Duke's goal following a smart move down the left flank, although a linesman's flag prevented Cotterill from letting rip.
Both sides fielded like-for-like 4-5-1 formations but the visitors seized possession amid a congested midfield and started to spray the ball around the park as Hull laboured.
But Bostock ensured there were signs of life in the Tigers as he crashed a shot just wide of De Vries' upright with a driven effort from 25 yards.
Cotterill continued to pose problems and his free-kick almost brought a goal as he curled one onto the angle.
But agonisingly for Rodgers, Hull took the lead moments later against the run of play.
Bostock - determined to make an impact on his debut - found a rare moment of time and space in midfield. He swivelled, carried the ball forward before unleashing a piledriver into the top corner that left De Vries grasping thin air.
Pearson's team now had the edge and began to impose themselves on Swansea, with Anthony Gardner somehow heading over from six yards from Richard Garcia's corner just before the interval.
Swansea needed to respond in the early stages of the second half but they made the worse possible start by falling further behind.
The industry of Garcia won his side a corner and when the Australian whipped the ball in, De Vries could only fumble the ball into the path of Ashbee.
The skipper made no mistake to lash home the volley with the net unattended to mark a memorable return to the side.
Rodgers made a double change to reinvigorate his team, bringing on Shefki Kuqi and Andrea Orlandi on the hour, and the latter nearly pulled one back when his header was saved by Duke.
But Hull held firm to earn a clean sheet and a well-deserved victory.