The Slovakian stunned the world number six with a 6-4 1-6 6-1 6-4 win, a result he has hailed as the best of his career.
"I was coming to the match to play a very solid game. I wanted to play really good tennis to play with the top star player," ESPN quotes Klizan as saying.
"I played very, very well today. I mean, I was concentrating each point and I won; I'm very happy that I did. He's like number six in the world and I beat him. I feel great. I don't know what to say. I'm very happy, and that's it.
"Third round, I think it's my best result in my career, and I'm looking forward to play Saturday third round. There is a chance to win again."