Elche, who have fallen quickly since being relegated from La Liga for unpaid tax debts in 2015, were resilient and by the end were matching Atletico blow for blow.
Thomas Partey sent the visitors in front with a downward header at the back post in the 17th minute after Jose Gimenez fired over a cross from near the corner flag.
Out-of-form striker Luciano Vietto saw an effort tipped on to the post from 15 yards out, while Elche goalkeeper Guillermo Vallejo pulled off a smart save to keep Fernando Torres at bay.
However, seven minutes after the break Lucas Hernandez brought down Lolo Pla in the area and the forward sent Miguel Angel Moya the wrong way from the penalty spot to level.
Vietto wasted two more good chances as Atletico failed to build the lead they wanted to take to the Wanda Metropolitano for the second leg at the end of November.
Elsewhere, third-division side Ponferradina stunned Villarreal 1-0 thanks to Sergio Cidoncha's first-half goal, leaving the top-flight side plenty of work to do in their home second leg.
Minnows Formentera held visitors Athletic Bilbao to a 1-1 home draw, while second-tier Valladolid went down 2-1 at home against Leganes thanks to an 88th-minute Claudio Beauvue strike.
John Guidetti's goal helped Celta Vigo to a 2-1 win at Eibar in an all-La Liga clash.
(Reporting by Rik Sharma,; Editing by Neville Dalton)
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