The visitors were the first team to threaten as they won a corner and Curtis Davies beat his marker to the ball and headed just wide of the target.
However, Everton were soon on top as Romelu Lukaku forced a good save from Allan McGregor, but from the resulting corner Kevin Mirallas gave the home side the lead. He side-footed the ball into the bottom corner beating an unsighted McGregor, who failed to move for the ball.
The Tigers came close to getting an equaliser through Abdoulaye Faye as he had two attempts from inside the box with headed efforts, but could not hit the target.
However, Steve Bruce's side were level when Yannick Sagbo converted from close range with an emphatic finish after Sone Aluko had done well to beat Gareth Barry down the right and found his teammate with a pinpoint cross into the area.
Hull should have been ahead just before half time when Faye again met a corner from Robbie Brady and his header was directed into the path of Aluko, who nodded it at goal but Leighton Baines was on the line to clear.
The visitors came out for the second half and continued their good play and could have taken the lead as Sagbo laid the ball off for Jake Livermore; his low effort looked to be creeping into the until Sylvain Distin diverted it away from goal.
Everton capitalised on this and regained the lead when Ross Barkley sent Mirallas down the left with a sublime through-ball and the Belgian cut it back to substitute Steven Pienaar, who guided his effort into the bottom corner.
Kone should have ended the game when he was set clear minutes later as he tried to beat McGregor, but the keeper stayed big and made a stop with his feet. However, his efforts were in vain as the Toffees held out for the win.