Second-placed Saints have now won 10 of their last 11 games in all competitions after Wanyama took full advantage of a blunder from Hull's third-choice goalkeeper to secure a 1-0 victory.
Jakupovic, starting only his third Barclays Premier League game in the absence of Allan McGregor and Steve Harper, scuffed a third-minute kick straight to the Saints midfielder and left himself hopelessly out of position.
The net was completely unguarded but it required a cool head and sound technique to find it from such range, a challenge Wanyama made light of as he stroked home his third of the season.
Hull's initial response was full of energy, with Mohamed Diame to the fore, but as the game progressed the home threat became increasingly blunt.
It was Southampton who came closest to adding another, the lively Steven Davis denied by Jakupovic from a poacher's effort and substitute Sadio Mane turning a glorious chance wide from point-blank range.
For England manager Roy Hodgson, making a first trip to the KC Stadium since taking the job, there was little outstanding from the six eligible players.
Fraser Forster and Nathaniel Clyne did nothing to play their way out of contention but team-mate Ryan Bertrand and the Hull hopefuls Tom Huddlestone, Curtis Davies and Jake Livermore did not demand attention.
The defining moment of the first half, and ultimately the match, came almost immediately.
Robbie Brady's back-pass left plenty to be desired but Jakupovic should still have been able to get the ball clear.
Instead he miskicked straight along the ground to Wanyama and left himself stranded.
The Kenyan still needed to find the net from his distant position and he finished with aplomb.
Hull rallied quickly, Brady giving Clyne a couple of scares on the left and Diame taking the midfield area by the scruff of the neck.
Diame gathered Davis' loose pass and dribbled into the area, exchanging a one-two with Abel Hernandez before seeing his drilled cross-shot cleared.
Hatem Ben Arfa briefly threatened too, working Forster from the edge of the box then jinking his way to the byline before Southampton smothered his cutback.
For all their energy Hull were struggling to battle past Jose Fonte and Toby Alderweireld, who marshalled the back four superbly under pressure.
The same could not be said of the Tigers defence, with Jakupovic's nerves seemingly spreading.
Davies almost slipped in possession under pressure from Dusan Tadic and a moment of hesitation from Paul McShane put former Hull striker Shane Long through.
The forward, a £12million summer signing, spared his fellow Irishman's blushes with a non-committal header.
In the 35th minute he had an even better chance against his old side, teed up by Davis' neat pass, only to see a combination of McShane and Jakupovic block his shot.
Hull needed to improve in the second half, Ben Arfa having gone quiet to leave Hernandez looking increasingly isolated.
Diame had a half-hearted penalty appeal waved away after Wanyama's powerful challenge just after the restart but Hull were otherwise looking uninspired in the final third.
The game was nearly put to bed in the 57th minute when Graziano Pelle made the slightest connection with an overhead kick, wrong-footing the Hull defence as the ball bobbled through.
Davis was alert, though, and looked certain to bury from close range when Jakupovic made a brave block.
Hull boss Steve Bruce made a much-needed change with 25 minutes to go, Sone Aluko on for McShane and the system switched from 3-5-2 to 4-4-2.
Southampton had already made their first substitution, Long hauled off for Mane after a modest showing on his old stomping ground.
Mane had a nearly unmissable chance to secure the points in the 70th minute, meeting Tadic's cross a couple of yards out in a central position.
But somehow he sent his shot into the ground and bouncing wide to widespread disbelief.
Pelle, unusually quiet by his own recent standards, should have been on the scoresheet in the 89th minute but slotted into the side-netting from six yards.
The vibrant away support had to make do with one goal despite their continued threat, Davis lashing the final chance into the stands.
Source : PA
Source: PA