Los Blancos made three changes from the win over Granada, as Luka Modric and Marco Asensio were both recalled alongside a rare start for left wing-back Marcelo.
The game got off to a quick start and Marcelo came very close to giving Madrid an early lead in the 13th minute with a powerful strike from the edge of the box that came off the underside of the crossbar.
Asensio also came very close for the visitors in the 20th minute with a dipping strike from range that cannoned onto the crossbar but Modric reacted the quickest from the rebound and powered home a header into a gaping net for the opening goal.
Zinedine Zidane's charges thought they had won a penalty with eight minutes left of the first stanza after the referee initially pointed to the spot for a handball by Ivan Marcone inside the box but after reviewing the incident on the Video Assistant Referee (VAR) monitor, he reversed his decision, as replays showed that the ball struck Marcone's knee.
Elche had good start in the second stanza after they were awarded a penalty seven minutes from the break for a shirt pull by Dani Carvajal on Antonio Barragan inside the box. Fidel brought the hosts back into the game with a low strike past the out-stretched arm of goalkeeper Thibaut Courtois from 12-yards.
The game was opening up after that, with Los Blancos having the momentum and shot-stopper Edgar Badia did well in the last 20 minutes to deny Carvajal and Sergio Ramos from Madrid's successive attacks.
Karim Benzema should have snatched the winner for Zidane's men in the last minute after he was left unmarked inside the six yard box to attack a cross by Modric but his jump lacked the necessary spring, as he brushed the lofted cross.
Elche almost won it with the last kick of the match from a free-kick by Gonzalo Verdu on the edge of the box but Courtois made a great save for the share of the spoils, as second-placed Madrid fell two points behind leaders Atletico Madrid, who have two games in hand.