The British number one came from a set down to beat the Croatian in four sets on Tuesday night and claims that he is happy to have met expectations in New York.
Murray said: "I've had a lot of tough matches - everyone goes through them. We've seen Rafa [Nadal], Novak [Djokovic] and Roger [Federer] struggle here.
"There is pressure on you to perform well, expectations are high and sometimes you're nervous.
"He played very well at the start and I struggled a little bit. When I managed to get the breaks back in the second set I relaxed a little bit."
Murray will face Tomas Berdych in the semi-finals after the Czech knocked out Roger Federer at Flushing Meadows.