Luka Modric scored his third league goal of the season to give Zinedine Zidane's men the lead after 20 minutes as Madrid attacked freely while leaving some dangerous gaps at the back.
Elche created numerous good chances before equalising from the penalty spot, Fidel converting after Dani Carvajal's blatant foul on Antonio Barragan.
The result put a dampener on the feel-good factor Zidane had managed to generate around his club after bouncing back from a slow start to the season, with Atletico two points clear at the summit having beaten Getafe 1-0 earlier on Thursday (AEDT).
Toni Kroos headed narrowly wide from 10 yards in the third minute as Madrid started brightly but it was not long before Thibaut Courtois – making his 100th appearance for the club – had to get down smartly to parry Emiliano Rigoni's fierce drive away from the corner of his net.
Play swung from end to end and Marcelo shook the crossbar with a rasping half-volley from the edge of the Elche box in the 13th minute.
Moments later Modric gave Madrid the lead, sending a header into the top-right corner after Edgar Badia palmed Marco Asensio's thunderous shot onto the crossbar.
Madrid was awarded a penalty only for VAR to show that the ball hit Ivan Marcone's leg and not his hand, and six minutes into the second half Elche won a spot-kick of their own when Carvajal pulled down Antonio Barragan.
Fidel buried the penalty into the bottom-left corner of the net and 10 minutes later Lucas Boye grazed the outside of the post with a low shot from the edge of the Madrid box.
Boye fired high and wide when he burst through the ranks of Madrid defenders again and Carvajal missed a chance to atone for his error when Badia saved his close-range effort on the counter.
The returning Eden Hazard missed the only opening that came his way and Carvajal conceded a stoppage-time free-kick from which Gonzalo Verdu drew a fine save from Courtois, who parried away the last kick of the game to ensure the points were shared.