The only goal of Saturday’s Premier League game arrived in the third minute, with third-choice goalkeeper Eldin Jakupovic enduring a moment to forget for the hosts.
His poor clearance fell kindly at the feet of Wanyama, who still had a lot to do from 40 yards out.
The Kenyan returned the ball with interest, though, into an unguarded net, with his composed finish sealing a 10th victory for Southampton in their last 11 outings in all competitions.
For England manager Roy Hodgson, making a first trip to the KC Stadium since taking the national team post, 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, Robbie Brady giving Clyne a couple of scares on the left and Mohamed Diame taking the midfield area by the scruff of the neck.
Diame gathered Steven 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.
Curtis 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 Sadio 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.
Phil Thompson’s view
Southampton thoroughly deserved it. In all departments they had the upper hand. The defenders did their jobs, the midfield was very good and was rotating and then there was the two players up front. Tadic was going left, right and long and stretching teams, while Pelle showed great hold-up play and proper centre-forward play. These were all positives and Ronald Koeman must be thinking this is easy. He has won seven out of the last eight Premier League games. It’s just astonishing what’s going on because this is a good team.
There’s no real weaknesses. The only thing I would say is they should have scored more goals. Steve Bruce will be disappointed. He started off with his 3-5-1-1 formation, but Hernandez was just isolated at home. Brady shined playing at left wing-back, but they had just one shot at home in the whole of the 90 minutes. Fraser Forster had a very easy afternoon.