A header from Luka Modric gave the champions the lead in the 20th minute but Elche found a way back into the game when Fidel Chaves levelled from the penalty spot early in the second half.
Both sides had chances to take all three points, with Elche forward Lucas Boye striking the post, while Real defenders Dani Carvajal and Sergio Ramos were thwarted by fine saves from the home side's keeper Edgar Badia.
The draw halted a six-match winning streak in all competitions for Real and left them second in the standings on 33 points, two behind city rivals Atletico who also have two games in hand. Elche are 15th on 16 points.
"We started well but our opponents had studied us well, they put everyone behind the ball and tried to hit us on the break and it's not a huge surprise we couldn't win," said Real midfielder Casemiro.
Real had pulled themselves back into the title race with a run of five league wins in a row before the Christmas break and looked likely to notch another after an excellent first half.
Zidane's side had a penalty incorrectly awarded for a block by an Elche player which the referee thought had struck his arm, but the official corrected the decision after reviewing it on the pitchside monitor.
Elche only troubled Courtois once in the first half but earned a golden chance to level when Carvajal was penalised for grappling with Antonio Barragan.
Courtois guessed the right way but Fidel's penalty had too much power.
They had their impressive Badia to thank for keeping it that way, the keeper denying Ramos from point-blank range, moments after flying across his goal to push away a shot from Carvajal.
Real's record signing Eden Hazard returned after over a month out injured but could not help his side get the winner they craved.
Courtois didn't agree with the penalty decision after the match.
"It seems a soft penalty to me, you get these little shirt pulls in the area, I didn't see it, but there are referees who like to give them. There was another on Benzema, if you don't give one, you can't give the other, you have to give both," Courtois said.
"They're things that happen in the area. But we can't say anything about the referee, we were more focused on the yellow cards and fouls than on our play."