Levante, who had lost five of their previous six games, dominated throughout and scored through Nano in the third minute and Felipe Caicedo in the 59th.
Goalkeeper David de Gea prevented a heavier defeat as Atletico, despite having Argentina's Sergio Aguero and Uruguay's Diego Forlan up front, struggled to create chances until late in the game.
"It's a defeat without any positives," said Sanchez Flores. "The first goal condemned us and the second was a public execution.
"We are going through difficult times, times of crisis and now what we have to do is manage our way through this crisis. We have gone from heaven to hell in a week and now we have to turn it round."
Atletico, who also lost to Espanyol last weekend, are seventh with 20 points from 14 games but are in danger of slipping further with their rivals still to play this weekend.
The defence of their Europa League title also hangs in the balance after a midweek defeat to Aris Salonika.
The Greeks are at home to bottom club Rosenborg Trondheim in their final match in two weeks when Atletico visit Group B leaders Bayer Leverkusen.
Aris only have to match Atletico's result to go through to the last 32.
