The tennis champion said he used to struggle to get out of bed but now relishes early starts so he can spend time with Sophia Olivia.
He told the Chris Evans Breakfast Show on BBC Radio 2 that he "always wanted to start a family".
The world number two was back on the practice court ahead of Great Britain's Davis Cup match with Japan next week.
Speaking about fatherhood, he said: "It's something I've been looking forward to for a long time so I'm sure it's going to have a lot of positive effects on my life.
"For me, the first time I was left alone with her was when it really sank in. It hits you that this is a huge responsibility, but during every day now I really look forward to going home to see her.
"Waking up is a lot easier now. I used to love sleeping in later but getting up early and getting to see her in the morning is great."
The Davis Cup marks Murray's return to a match court for the first time since losing his fifth Australian Open final to Novak Djokovic.
Immediately after the defeat, he paid an emotional tribute to his wife Kim before catching the first flight home for the birth of his daughter.
The top four players in the world are now fathers and Djokovic has credited the birth of his son Stefan as one factor behind his success.
Murray was reticent about whether he thinks his daughter, who was born on 7 February, will change his game.
He said: "I think obviously priorities change significantly.
"I haven't played a competition so I don't know how I'm going to feel the next time I'm on a match court, but I always wanted to start a family."
New father Andy Murray has revealed that his baby daughter is making it easier for him to get up early.