Madrid collected a third consecutive Champions League crown after substitute Gareth Bale struck twice from the bench in Saturday's 3-1 triumph.
But Karius' errors had a huge impact on the result, the German goalkeeper bizarrely throwing the ball in off striker Karim Benzema for the opening goal in Kiev.
After Bale's brilliant bicycle-kick restored the Madrid lead following Sadio Mane's leveller, Karius then allowed a long-range drive from the Wales international to escape his grasp, wrapping up the title for Zinedine Zidane's men.
"I don't feel anything right now. Today I lost my team the game," Karius said to talkSPORT.
"I feel sorry for everyone. I'm sorry for everyone, for the team, for the whole club. The mistakes cost us dearly.
"If I could go back in time I would. I just feel sorry for my team. I know I let them down today.
"It is very hard right now, but that is the life of a goalkeeper. You have got get your head up again.
"For now, I'm just really sorry for everyone that those goals cost us the title basically."