The 25-year-old midfielder, who signed from Aston Villa after making a U-turn over his switch to City last week, lasted less than 20 minutes after a tussle with Luka Modric resulted in him being carried from the pitch on a stretcher.
City will now face an anxious wait to determine the severity of Delph's injury, with their opening Barclays Premier League match against West Brom just over two weeks away.
Moments after Delph had been replaced by Jesus Navas, Real added insult to injury as an incisive move resulted in Karim Benzema volleying the opening goal past City goalkeeper Joe Hart.
Cristiano Ronaldo added the second shortly after before Pepe made it three from a header.
Yaya Toure pulled one back for City in first-half stoppage time from the penalty spot but Real winger Denis Cheryshev added Real's fourth with 15 minutes to go and Manuel Pellegrini's side never looked like launching a comeback.
After an evenly-matched start, City were dealt a double blow after Delph went down following a challenge with Modric and Gareth Bale then set up Benzema from the right wing, the French striker firing past Hart from the middle of the box in the 21st minute.
Ronaldo quickly doubled Real's advantage when Toni Kroos launched a perfectly-weighted ball to his feet before the Portuguese wriggled clear of Aleksandar Kolarov to chip into the net
Hart parried the ball but it went backwards and over the line, with his scrambled efforts to scoop it clear in vain.
The Spanish giants moved 3-0 up just before the break from the set-piece when Isco sent in a corner and found Pepe, who climbed the highest in the box from 10 yards out to power a header past the motionless Hart.
Toure pulled one back for City in first-half stoppage time from the penalty spot after Sergio Ramos handled the ball following a flick-on from Raheem Sterling.
After the break, Real went close twice in quick succession around the hour mark when Ronaldo almost played in Isco with a one-two before the Portuguese playmaker burst into the box and fancy footwork almost undid the City defence with a shot that crept wide.
Real did add a fourth in the 73rd minute through Cheryshev, who bundled in the ball as it fell at his feet from close range following a surge into the box by Isco, with City unable to gain further traction.
Source : PA
Source: PA