Third-choice Hull keeper Jakupovic continued to deputise in the absence of Allan McGregor and Steve Harper in Saturday's game at the KC Stadium.
And his mishit clearance after just three minutes fell into the path of Wanyama, who cleverly guided the ball home from distance for the only goal of the game.
On balance of play it was a deserved three points for Southampton, who remain second in the table, as Jakupovic's opposite number Fraser Forster was rarely tested in his bid to impress the watching England manager Roy Hodgson.
Hull, who made one change from last week's 0-0 draw at Liverpool with Paul McShane coming in for Alex Bruce, failed to stamp their authority on the game and fell to a second home league defeat of the season.
Wanyama's early strike gave in-form Southampton a perfect start, and former Hull striker Shane Long - who moved to St Mary's in August - was a constant threat for the visitors in the first half.
The second period failed to produce much in terms of goalmouth action, but Southampton substitute Saido Mane should have made it 2-0 when he put a gilt-edged opportunity from inside the six-yard box over the crossbar with 20 minutes remaining.
However, Wanyama's third Premier League goal of the season proved enough for Southampton, who have now won 10 of their last 11 matches in all competitions.
Following the Kenyan's opener, Hull enjoyed a spell of pressure and Hatem Ben Arfa should have done better when he stot tamely at Forster from the edge of the box on his favoured left foot.
But at the other end, Southampton nearly benefitted from another Jakupovic error.
The Pole hesitated after realising Long would reach a long ball first and was relieved the ex-Hull man could not direct his looping header into the goal.
Southampton looked assured in possession after the break as Hull tried desperately to get a foothold in the game.
The visitors almost had a second in fortuitous circumstances when Graziano Pelle's acrobatic overhead kick fell kindly for Steven Davis, who could only half-volley straight at Jakupovic from close range.
Hull's Mohamed Diame was perhaps lucky to escape punishment for a crude challenge on the heel of Dusan Tadic.
But Tadic picked himself up and his teasing cross found Mane in a great position in the penalty area, only for the forward to flick his shot into the ground and over the crossbar in the 70th minute.
The hosts had tame shouts for a penalty denied when Robbie Brady's cross appeared to hit Tadic on the hand, but Steve Bruce's side rarely threatened the Southampton goal in the closing stages.
Indeed, it was the visitors who went closes to scoring again, as Pelle wasted a great chance to round off the win in the 89th minute when he clipped a shot wide at the near post after a fine delivery from Mane.
It mattered little, though, as Ronald Koeman's men march on.
