The powerful Ivorian striker stole the ball and galloped away to set up Jose Javier Barkero for the opener in the 20th minute, while Juanlu’s shot was chested over the line by Valdo just after the break.
Kone broke clear to make it 3-0 soon after and then pounced on a rebound as Asier del Horno’s free kick came back off the woodwork just after the hour mark.
Last month, tiny Levante hit the top of La Liga for the first time since being founded 102 years ago but had slid off the pace with three consecutive defeats.
The Valencia-based club stayed fourth with 26 points from 13 games, eight behind leaders Real, who thumped nine-man Atletico Madrid 4-1 in the city derby on Saturday.
Jose Mourinho’s side moved six clear of second-placed Barca, who suffered a shock first defeat of the campaign at Getafe where they lost 1-0.
Valencia are third with 27 points after a 2-1 win at Rayo Vallecano.
Earlier, Real Sociedad defender Inigo Martinez scored from the halfway line for the second time this season to earn his side a last-minute winner at Real Betis in a 3-2 victory.
Martinez caught Betis keeper Casto Espinosa off his line with a long-range shot from wide on the left, having scored a similar effort against Athletic Bilbao in the Basque derby last month.
A first win in nine matches eased the pressure on Sociedad coach Philippe Montanier but piled it on his opposite number Pepe Mel, who is without a win in nine.