A Matias Dituro own goal handed the visitors a 25th minute lead at Estadio Manuel Martinez Valero.
Madrid were two goals to the good when Kylian Mbappe fired home after 33 minutes.
The second half seemingly belonged to Martin Anselmi's charges as Abiel Osorio pulled a goal back with 71 minutes on the clock before Fer Nino's 83rd minute strike to level the score.
But Espi replaced Trent Alexander-Arnold after 86 minutes and netted a 91st minute winner to leave Jose Mourinho's men in second place on the league standings with 15 points from six games- level with leaders Barcelona but from a game more.
Elche are bottom of the table, with this point just their second draw of the season alongside four losses.
Speaking after the game, Mourinho said on Roundtable: "It seems obvious to me. It is easy to understand. There are finished matches in which we have chances to kill, and we do not kill.
"After not killing, we go from ease to not controlling. We have not scored. When everything seems easy, you leave things undone. The explanation I find is that if the match seems easy and you do not finish it, then you drop it; you lose control; you misinterpret the ease.
"It can happen. Nothing is easy. The players need to rest; tomorrow they have a day off. If we play badly, from the first to the 90th, the worry is gigantic.
"If we have top periods and others are not proactive, without expectation, without intensity. Of course, it worries. But we won."
RB Leipzig managing director for sport Marcel Schafer has admitted the club would have preferred to keep Yan Diomande, despite ultimately sanctioning the winger's record-breaking move to Real Madrid.
Jose Mourinho has praised Arda Guler's growing influence at Real Madrid and believes the young midfielder has the potential to become a major player for the club.
Kylian Mbappe bagged a brace as Real Madrid thrashed Rayo Vallecano 4-1 to move level with Barcelona at the top of LaLiga's standings.
Kylian Mbappe has backed himself to win the Ballon d'Or after a remarkable individual season, insisting there was no player better than him despite Real Madrid and France falling short of major honours.
