The Reds had nine shots with Salah on the pitch and none for the rest of the half.
Real Madrid scored their opener at 51 minutes after a gaffe from Reds keeper Loris Karius allowed Karim Benzema to score.
Liverpool's only goal came when winger Sadio Mane netted four minutes later to equalise.
Bale had only been on the pitch two minutes before he rose in the air to execute a stunning bicycle overhead kick that found the top corner.
His goal put Real Madrid 2-1 up after 64 minutes.
Karius suffered further humiliation when Bale's 83rd minute long-range strike slipped through his hands to deny Liverpool their sixth European cup.
Real Madrid's victory in Kiev marks the third season in a row they have secured European club football's biggest prize.
The Spanish giants have sealed their 13th European Cup, their fourth in five seasons.
Liverpool manager Jurgen Klopp's long wait for a European trophy continues, after the German lost in the 2013 final as manager of Borussia Dortmund.