The five-time champion is in the unusual position of being only the seventh seed after his worst season for more than a decade.
Federer has won only one title, reached one grand slam semi-final and seen his remarkable record of consecutive slam quarter-finals ended at 36 with a second-round defeat by Sergiy Stakhovsky at Wimbledon.
A run at the US Open is much needed, and Federer made a steady if unspectacular start with a 6-3 6-2 7-5 victory over Slovenian Grega Zemlja on Arthur Ashe.
The 32-year-old said: "Clearly when you win everything, it's fun. That doesn't necessarily mean you love the game more. You just like winning, being on the front page, lifting trophies, doing comfortable press conferences. It's nice.
"But that doesn't mean you really actually love it. That maybe shines through more in times when you don't play that well. For me, I knew it, winning or losing, practice court or match court, that I love it.
"Clearly when I had my two girls, I also wasn't sure right off the bat how it was going to be after that. Was I going to be able to play the same schedule? Was my love for the game as big? Were we going to be able to cope with the whole thing or not?
"(We) managed it totally fine. They were at the court today [Tuesday]. I'm so happy to see them before and after the match. I'm in a good spot right now. I want to enjoy it as long as it lasts."
The match had been scheduled for the night session on Monday but rain meant it was held over, and Federer, who next meets Argentina's Carlos Berlocq, was happy with how he handled the situation.
He said: "There wasn't much rhythm out there, but I was happy the way I played overall. It's a first round after all. There was a bit of a breeze. I should have played at night., I played during the day. Overall I'm pleased to be in the next round."