The tiny Valencia-based club went top of the table earlier this season for the first time since they were founded 102 years ago but a run of three defeats meant they lost ground.
Levante are now fourth with 26 points from 13 games, eight behind leaders Real Madrid who thumped Atletico Madrid 4-1 on Saturday.
Jose Mourinho's side moved six points clear of second-placed Barcelona who suffered their first defeat of the campaign at Getafe, losing 1-0. Valencia are third with 27 points after a 2-1 win at Rayo Vallecano on Saturday.
Ivorian striker Kone stole the ball and galloped away to set up Jose Javier Barkero for Levante's opener in the 20th minute and then Juanlu's shot was chested over the line by Valdo just after the break.
Kone broke clear to make it 3-0 for the home team before pouncing on a rebound in the 62nd minute as Asier del Horno's free-kick came back off the woodwork.
"To have 26 points at this stage of the season is the best news for us from today," Levante coach Juan Ignacio Martinez told reporters.
"We are the smallest team in the division and the fact we are two points behind Barca means little. We just need to accumulate as many points as possible."
Promoted Granada upset in-form Athletic Bilbao 1-0 at San Mames, a first-half header from Inigo Lopez ending an unbeaten run of 11 matches in all competitions for Marcelo Bielsa's side.
