Mohamed Salah's early penalty put Liverpool on track and Divock Origi came off the bench to seal victory with three minutes to go, as the Reds celebrated a sixth European triumph.
The Reds earned redemption for their painful 3-1 loss in last year's final against Real Madrid in Kiev, when Karius was badly at fault for two goals.
The goalkeeper, who joined Besiktas on loan ahead of this season after the signing of Alisson from Roma, was quick to congratulate his parent club.
Posting on Twitter, he wrote: "Congrats, @LFC. Really happy for everyone at the club and fans. You deserved this."