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The Saints had the perfect start to the match when Eldin Jakupovic's clearance fell straight to Wanyama, and with the keeper out of his goal the midfielder fired an effort from 45 yards which found the back of the net.
Steve Bruce's side managed to get a foothold in the game as they dominated possession, but it was the visitors who nearly doubled their lead. Shane Long chased a long ball and Jakupovic was hesitant in coming out, but fortunately for the keeper the striker's effort went wide.
Long came close again for the Saints when he latched on to a pass from Steven Davis. He decided to try to poke the ball past Jakupovic, but the keeper made a good stop to keep his side in the match.
In the second half, Southampton came close to extending their lead when Graziano Pelle tried an ambitious overhead kick in the box, which was deflected into the path of Davis. The midfielder tried to lift the ball over Jakupovic, but the keeper made another solid stop.
Sadio Mane should have put the game beyond doubt for the visitors when he managed to beat his man at the near post to collect Dusan Tadic's cross, but he failed to he somehow failed to hit the target from close range.
Hull did their utmost to engineer a way back into the game, but for all their possession they could not find a way past the resolute Southampton backline, who absorbed everything thrown at them with ease.