Real were denied the chance to put more breathing space between themselves and Sevilla, who are four points behind in second. Eder Militao spared the home side's blushes with a 92nd minute header but Real will consider this two points dropped rather than one gained.
Sevilla also drew on Saturday and Real Madrid coach Ancelotti said: "I think Sevilla are thinking the same thing, an opportunity missed. When you don't win, you miss an opportunity. It was quite a crazy Saturday and Sunday.
"Sevilla drew, us too. Atletico won in the end, a game that looked lost. It's a very competitive league. Elche play good football, they're well organised with and without the ball. To win you have to compete."
Madrid pushed for an equaliser throughout the second half and were frustrated by Elche's resolute defending. Pere Milla hit them on the counter attack with 14 minutes to go before Luka Modric reduced the deficit with a penalty of his own before Militao's late, late equaliser.
Manager Carlo Ancelotti made seven changes to the team that eliminated Elche from the Copa del Rey, with midweek matchwinner Eden Hazard starting on the right wing.
Vinicius Junior got in behind the defence in the 11th minute but was unable to beat goalkeeper Edgar Badia, and Benzema and Casemiro were both denied before Madrid were awarded a penalty when Vinicius was fouled by Helibelton Palacios.
Benzema blasted the spot kick over the bar -- his first penalty miss as a Real Madrid player -- and then Elche took the lead on 42 minutes, Boye rising unmarked to head Fidel's cross past Thibaut Courtois.
After the break, the home side were awarded another penalty when Hazard went to ground inside the box, but referee Ricardo de Burgos Bengoetxea changed his mind after consulting the pitchside VAR monitor.
Ancelotti introduced substitutes Luka Jovic -- who replaced the injured Karim Benzema -- and Rodrygo as Madrid kept Elche pinned back, before Milla's 76th minute strike, shooting low into the bottom corner.
Madrid were awarded another penalty six minutes later, this time for handball, and Modric scored to set up a frantic final period before Militao stole a point late on.
"I think in the end we have to go home happy, because we were 2-0 down, and once again the team showed a lot of courage, the will to win," Real Madrid's Lucas Vazquez said after the match.
"It's true that we weren't quite clinical enough in the final third throughout the game, I think we created enough chances to get the three points, but seeing how the game went, we have to be happy with how the team looked, getting the draw in the end."