Vinicius Junior proved to be Real's hero on the night, scoring the only goal of the game fifteen minutes into the second half.
Liverpool started the brighter of the two sides, carving out a number of efforts on target but Thibaut Courtois in the Real Madrid goal was equal to the task on all occasions.
The Belgium goalkeeper showed good reflexes to divert Mohamed Salah's 16th minute effort away from goal.
Real Madrid took the lead on the hour mark after Federico Valverde drilled a ball across the Liverpool penalty box, leaving Vinicius Junior to tuck in unchallenged at the far post.
Courtois again came to Real Madrid's rescue when he tipped a fierce Sadio Mane strike against the post in the 20th minute.
Salah sought an immediate response for Liverpool, testing Courtois from range in the 62nd minute but his effort was punched clear.
Salah called Courtois into action again with eight minutes left but when all looked set and done, the former Chelsea goalkeeper threw out an arm to divert the ball over the bar.
