Rodri scored either side of a Sebastian Dubarbier own goal as the home side looked like they were heading for victory with the game entering the last 10 minutes.
But Giovani dos Santos levelled in the 84th minute on his top-flight debut for the Yellow Submarine before setting up Jonathan Pereira to complete a stunning turnaround with three minutes remaining.
Elche's first La Liga appearance in 24 years ended in disappointment as they were comprehensively beaten 3-0 away to Rayo Vallecano.
Fran Escriba's men returned to the Spanish top flight after being crowned Segunda Division champions last season.
But Monday's comeback ended in disappointment as an Alberto Perea strike was sandwiched by an Alberto Bueno double at the Estadio del Rayo Vallecano.
The match was notable for a silent protest held at the start by the Vallecano supporters in a campaign against the league's fixture scheduling policy.
The home fans, though, were in fine voice by the time Bueno opened the scoring in the 40th minute when he timed his run to perfection before drilling into the bottom corner.
Perea made it two with a spectacular long-range strike a minute before half-time and Bueno matched that with a 25-yard effort in the 74th minute.
Meanwhile, Celta Vigo surrendered a 2-0 lead as Espanyol fought back to claim a 2-2 draw at the Estadio de Balaidos.
Alex Lopez and Charles were the beneficiaries of assists from Michael Krohn-Dehli either side of the half-time whistle.
But Espanyol responded with goals from Victor Sanchez and Thievy Koulossa to earn a share of the spoils.