The hosts were soon back on the attack and called Keylor Navas into action for the first time in the 18th minute when Suarez tried to curl the ball in from the left-hand side of the box, but he could not beat the Levante keeper.
Levante had a chance to break the deadlock when Cisma brought down Ruben Garca, giving the visitors a free kick from a dangerous position, but Hector Rodas could only head over from Pape Diop's delivery.
However, it was the home side that edged ahead with just over an hour gone when Rodrigues found Richmond Boakye, who drilled the ball in low.
Their lead lasted just over 10 minutes though, as Angel had a shot put behind for a corner before tapping in from the delivery back into the box.
Rodrigues had a chance to restore Elche's lead following a long ball forward from Sanchez, but he volleyed his finish just wide.
Carlos Gil tried to force a winner in stoppage time with a shot from outside the box, but it finished well off target, ensuring that the sides finished level.