After going two seasons without a major trophy, Madrid appointed veteran Mourinho for a second spell in charge and made a slate of signings this summer to overhaul the squad.
French superstar Mbappe, playing under Mourinho for the first time, intercepted a poor back pass and fired home in the sixth minute to send the 15-time European champions ahead.
Dean Huijsen headed home the second from Arda Guler's corner as Madrid took full control against their German opponents.
Half-time substitute Carlos Espi nodded home Madrid's third in the second half, before 140 million euro Ivory Coast winger Diomande and England international Jude Bellingham also joined the fray from the bench.
Barca thrash Basel as Bisiwu bags brace
Elsewhere, Spanish champions Barcelona recorded an entertaining 5-2 victory at Basel in a friendly as summer signing Anthony Gordon played for the first time since his switch from Newcastle.
Another new arrival, Karim Adeyemi, put the Catalans ahead after being teed up by Raphinha, but youngster Jean Doucoure slammed home before the break to level for the Swiss.
Egyptian forward Hamza Abdelkarim restored Barca's advantage early in the second half after Adeyemi fed him.
Lamine Yamal, making his first appearance since winning the World Cup with Spain, tucked home a penalty for the third after Jules Kounde was fouled.
Kazeem Olaigbe netted from the spot for Basel before Barca substitute Jesse Bisiwu, signed from Club Brugge, hit a brace in a late but electric cameo.
Barcelona have one more friendly to play, at home against Al Ahly on Wednesday, before their LaLiga season begins at Elche on August 23rd.
