Madrid looked to assert their authority from the first whistle and should have gone ahead inside two minutes, but Vinicius Junior scuffed his effort wide from six yards out.
Elche responded well and Tete Morente dragged a shot past the far post, although the hosts were grateful when Dean Huijsen – who scored in both H2Hs last season and was one of five changes to Los Blancos’ starting lineup – headed straight at Matias Dituro after rising high to meet a corner.
The chances continued at both ends as the half wore on, with Abiel Osorio lashing over the crossbar on his first start for Elche before Vinícius poked another presentable chance wide.
It seemed like a matter of time until the opening goal would come, but it arrived in slightly fortuitous circumstances as Arda Guler’s 25th-minute free-kick cannoned off the far post and took a flick off Dituro’s glove, diverting the ball into the net for an own goal.
Kylian Mbappe doubled the visitors’ lead eight minutes later when he swept home Yan Diomande's cross for his seventh La Liga goal of the season, and his first on the road in the league since February.
The striker saw another goal ruled out for offside shortly after before Thibaut Courtois made a sharp low save from Morente in stoppage time, as Elche went in behind at half time for the fifth time this season.
To their credit, Los Franjiverdes’ heads did not drop, and the lively Osorio saw a close-range effort deflect behind off Ibrahima Konate shortly after the restart.
Morente then had Courtois scrambling with a long-range shot that flew just wide, but Elche were always vulnerable on the break and Vinicius should have sealed victory with one such attack, only to drag his effort past the post.
The Brazilian was left to rue that missed opportunity 19 minutes from time, when Osorio headed home Lucas Cepeda’s cross to score his first Elche goal.
That set up a grandstand finish and Elche got the equaliser their second-half performance deserved seven minutes from time, as substitute Fer Nino sent a low shot into the bottom corner that gave Courtois no chance.
Madrid poured forward in search of a winner, and they found one in the first minute of stoppage time as Mbappe broke clear and squared the ball for Espi to finish into an empty net.
The victory puts Madrid behind Barcelona only on goal difference, while extending their unbeaten run against Elche to 20 games (W16, D4).
Meanwhile, Martin Anselmi is still awaiting his first win as manager of Los Franjiverdes, who are now bottom of La Liga after six matches (D2, L4).
Flashscore Man of the Match: Kylian Mbappe (Real Madrid)