Vinicius Junior scored just before the hour with Real's first attempt on goal and their goalkeeper Thibaut Courtois made several fine saves to deny Liverpool, who end the season with two domestic cups having also failed to win the Premier League.
In a battle of European heavyweights, Juergen Klopp's side had more chances but lacked sharpness when it mattered as Carlo Ancelotti became the first coach to enjoy four European Cup triumphs having won the trophy with AC Milan and Real twice, leapfrogging Liverpool's Bob Paisley and Real's Zinedine Zidane.
"I am a record man...