Kenyan midfielder Wanyama was the beneficiary of Jakupovic’s blunder as he finished coolly from 35 yards after a poor clearance from the keeper in the third minute at the KC Stadium.
That was enough to end Hull’s three-match unbeaten run and means Ronald Koeman’s surprising Saints will finish the weekend riding high in second place in the Premier League.
Jakupovic was the hero at Anfield last week as Hull held on for a goalless draw against Liverpool, but he was embarrassed early in his first start in the Premier League.
The Swiss keeper came to the corner of his penalty area to clear Robbie Brady’s back-pass, but the kick was poor and the ball fell to Wanyama, whose first time shot from 35 yards floated into the unguarded net.
Southampton have won every match in which they have scored first this season, yet Hull’s response to falling behind was spirited, with Mohamed Diame at the heart of their best attacking play.
The match was watched by England coach Roy Hodgson, whose side face Slovenia in a Euro 2016 qualifier on November 15 before a friendly match with Scotland three days later.
The Southampton starting line-up featured a trio of players hoping to get the call from Hodgson in goalkeeper Fraser Forster and full-backs Nathaniel Clyne and Ryan Bertrand, while Hull’s Tom Huddlestone and Jake Livermore have both been selected in the past.
Bertrand was well positioned to clear a dangerous cross from Diame after the Senegal midfielder exchanged passes with Abel Hernandez before Toby Alderweireld hooked the ball over his own bar following Ahmed Elmohamady’s cross.
Shane Long was handed his fifth league start of the season to face the side he left in the close-season.
Jakupovic went some way to atoning for his error in the latter stages of the first half when he prevented former team-mate Long from doubling the visitors’ advantage.
A cunning pass from Steve Davis sent the Ireland striker through on goal but Jakupovic was out quickly and courageously to smother the shot.
The Hull goalkeeper was called into action again 12 minutes into the second half, when he again reacted quickly, this time to push the ball way from Davis, who was poised to prod the ball home from six yards after Graziano Pelle’s overhead kick.
Jakupovic had earned a change in fortunes and he got some luck with 20 minutes remaining when Saido Mane contrived to miss a glorious chance from inside the six yard box.
Dusan Tadic’s low cross from the right was perfect but Mane made poor contact with the ball and his first-time shot looped over Jakupovic’s bar.
Fortunately for Southampton, that miss didn’t come back to haunt them and they held on to seal the points.