Carlos Espí came off the bench to score a late winner for Madrid on his debut for the Spanish giants.
The 21-year-old Espí was one of the standouts of La Liga last season when with Levante.
Espí was brought on in the 79th minute and scored 11 minutes later, pouncing after Kylian Mbappé had been knocked off the ball and firing past Marko Dmitrovic.
Jude Bellingham headed in a free kick in the ninth minute and almost scored in similar fashion early in the second half.
Álex Calatrava had scored a deserved equaliser for the Espanyol in the 30th minute, sweeping the ball past Thibaut Courtois.
Mourinho was hired in June, back for a second stint with Madrid some 13 years after he left the club. He was in charge from 2010-13, winning a Spanish league and a Copa del Rey.
It was Madrid's first match of the season as they and defending champions Barcelona sat out the first weekend because they had players return later from the World Cup.
In Italy, Inter Milan kicked off their title defence with a dominant 4-1 win over newly-promoted Monza in Serie A .
Inter forward Pio Esposito had a breakout season last campaign and the 21-year-old started right where he left off.
An impressive backheel flick from Esposito put the hosts 3-1 up in early in the second half, either side of Inter goals from Piotr Zieliński and Yann Bisseck.
Inter - who dominated Italian soccer last season, winning both the league and the cup in Cristian Chivu's first year in charge - got off to a great start when a fierce long-range shot from Hakan Çalhanoğlu gave them the lead in the sixth minute.
But sloppy defending for Monza's equaliser showed just why Inter signed England defenders Djed Spence and John Stones in the offseason.
Stones came on late on while Spence was in the stands along with England teammate Curtis Jones, who completed his move to Inter on Friday.
Last year's runners-up, Napoli, won 2-0 at Genoa thanks to two late goals, from Kevin De Bruyne and Antonio Vergara.
Como - who surprised with a fourth-place finish last season - were held to a 1-1 draw at Udinese.
in France, last season's runner-up Lens comfortably beat Auxerre 5-2
Auxerre gave away two penalties and also played most of the match with 10 men after defender Christ Makosso was shown a straight red card for a foul on Abadallah Sima in the 35th minute.
Florian Thauvin converted both penalties – the second one at the second time of asking after Auxerre goalkeeper Mamadou Diop saved the initial attempt but was off his line.
All of Lens goals, apart from Thauvin's first 12th-minute penalty, came in the second half.
11°C