Jude Bellingham headed Real Madrid ahead inside nine minutes, only for Fernando Calatrava to level for Espanyol before the half-hour, and it took a stoppage-time strike from debutant Espí, pouncing after Kylian Mbappé's driving run broke down in the box, to spare Mourinho an unwanted repeat of his previous LaLiga bow, a goalless draw with Mallorca in 2010.
Espanyol boss Manolo González said his side had stuck largely to their game plan until fatigue set in late on.
"Up until minute 70 it was what we'd worked on, then not so much. I think we defended them pretty well. I think the goals came from two errors of ours, and their dangerous moments came from their talent and Mbappé's breaks.
"Overall we defended and attacked well, until the tank ran empty."
González felt there had been more positives than negatives despite the defeat.
"In the second half it's normal that they push you, but we had good moments. We have to keep going, I think there's more positive than negative. Against these sides it's sometimes hard to get into the game because they demand a lot of you. I really liked that after the 0-1 the team got up, competed and played very well. That's what I wanted. We reacted with football. I'm very happy with the players."
He was critical of the officiating, though careful to stress it hadn't decided the outcome.
"The referee never conditioned the game, what bothers me are the criteria. Every time we took a throw-in, they counted it against us, never against them. On fouls where we broke forward it was the same, they didn't whistle for us. It didn't decide the game, but the criteria should be a bit more normal. When something is a foul one way and not the other, it wears you down."
Mourinho, meanwhile, said his side deserved the win but had no complaints about the difficulty of the contest.
"We deserved to win; Espanyol didn't deserve to lose. That's how it is. We had a lot of control in the second half, but they did things very well, right to the limit. It's tough for them to end up with nothing. You have to factor in physical condition — they didn't have World Cup players.
"They've been working together for weeks. We're missing minutes, kilometres. It's a win with soul, from the bench. A frustrated bench doesn't help, one that's fully invested gives you a lot."
Asked about Vinicius Júnior, who again drew heavy attention from opposition defenders, Mourinho was protective of his forward.
"I run the risk that you'll go back months and dig out what I said when I was up against Vini. Now Vini is mine. He's no saint, but what they do to him is too much. It's something that has a pattern, that repeats from one opponent to the next.
"I foul him, now you foul him. If I get a yellow, you foul him. He has to have a lot of emotional control. I've told him that if he scores, he should think carefully about how he celebrates it. We're in a social era of anti-bullying, and with Vini we're still in the bullying, from opponents and from the crowd. We have to educate him and prepare him as much as possible, and that's it."
On the performance of debutant Carlos Espí, who joined this summer and struck the winner, Mourinho pointed to the impact of a bench he had trusted to change the game late on.