The Ivorian striker cancelled out a first half goal from Jonas to equalise in the 53rd minute, to keep Juan Martinez's men within three points of their third placed opponents.
A disappointing draw at home will inevitably pile more pressure on the underfire Unai Emery, who faces a fight to keep his job with Malaga, Levante, Osasuna and Atletico Madrid all breathing down his team's neck.
"A draw is a bad result and we are not satisfied," Emery said. "The fans have protested the way they did but they supported us during the game.
"I am not worried about dropping out of the Champions League places but it us up to me to change this dynamic and find solutions," he said, adding that club president Manuel Llorente had spoken to him after the game saying the team had to improve.
The home side dominated possession from start to finish, but misplaced passing, niggly fouls and a litany of bookings disrupted the rhythm of the game.
The Valencia players were clearly feeling the pressure as they tried to force the killer pass against a very well organised Levante.
But, eventually the pressure told in the 35th minute. Artiz Aduriz played a fantastic reverse pass to Sofiane Feghouli inside the right back and the French midfielder slid the ball across to Jonas to tap the ball into an empty net.
The home side continued to control the game in the second half, but were caught dozing at the back six minutes after the interval.
Jose Barkero found Valdo at the back post with a lofted cross, and the midfielder nodded the ball down for Kone to sweep home.
A flurry of chances came in the closing stages with both sides guilty of missing easy chances. Kone squandered a great opportunity from close range to nick it for Levante.
And Valencia's Paco Alcacer headed straight into the hands of Gustavo Munua when he should have done better. Neither side managed to score a winner and Emery was forced to face the white hankies at the full-time whistle.
In the other games, struggling