Jude Bellingham got Los Blancos off to a fast start, heading home from an Arda Guler free kick to put them ahead in just the ninth minute.
But they were pegged back around the half-hour mark when Javier Hernandez created space for a cross from the left flank before playing a lovely ball into Alex Calatrava, who finished well to make it 1-1.
Madrid dominated large portions of the match, especially in the second half, earning over 60% of the possession and producing 21 shots to Espanyol's eight.
However, they were a bit wasteful with many of their chances, only getting 10 on target, and Marko Dmitrovic had a good game in goal for the hosts, making eight saves to keep the visitors at bay for much of the second half.
That is until stoppage time at the end of the game when Carlos Espi picked up the loose ball at the edge of the six-yard area after a scrappy piece of play and took it past the on-rushing Dmitrovic before sliding the ball into an empty net to secure the three points for Real Madrid.
Elsewhere in LaLiga over the weekend, Rayo Vallecano and Alaves played out to a 1-1 draw in front of a sparse crowd at Estadio Butarque.
A goalless first half saw a Rayo goal by Pelayo Fernández disallowed by VAR for handball.
Sergio Camello put the hosts ahead early in the second half, but substitute Mariano Díaz equalised late for Alaves with a strike from outside the box to share the points.
At a packed Estadio de la Cartuja before nearly 60 000 fans, Betis ground out a 1-0 win over Real Sociedad.
Rodrigo Riquelme scored the decisive goal just after the hour mark following a fine assist from Cucho Hernandez.
Real Sociedad pushed hard late on, hitting the post and forcing good saves from Alvaro Valles, but Betis held firm for their first win of the season.
It was a tough home debut for new Athletic head coach Edin Terzic at San Mames as they were beaten 3-1 by Sevilla.
Left-back Yuri Berchiche was sent off around the 11th minute to leave them down to 10 men for the vast majority of the game, and Sevilla quickly capitalised with Oso opening the scoring shortly afterwards.
After the break, substitutes Chidera Ejuke and Isaac Romero added further goals, exploiting defensive errors from the hosts. Aitor Paredes headed a late consolation for the hosts, but Sevilla claimed a comfortable victory.
A low-quality, cautious affair at Mestalla lacked clear-cut chances and resulted in a 0-0 draw between Valencia and Celta de Vigo.
Both sides struggled to create meaningful opportunities, with Celta looking slightly more organised and Valencia appearing somewhat disjointed throughout.
