Second seed Federer set up a second-round match against Sam Groth by wining 6-3 6-4 7-6 (7-4) at Flushing Meadows on Tuesday.
The 33-year-old Swiss hit 41 winners and fired down 10 aces to progress in the final grand slam of the year, but it was not all plain sailing as Australian Matosevic secured a break in the third set to set up a tie-break.
And Federer had not envisaged needing a breaker to secure his passage into the next round.
"It was a bit more difficult than I thought it was going to be in the third set," said Federer.
"Marinko has a big game."
Federer was watched by basketball legend Michael Jordan and the Swiss spoke of his admiration for the former Chicago Bulls star, who sat in his box during the game.
"I was so happy that he came into the world of tennis for a little bit and that he got to see me as the character in the whole show was pretty cool," added Federer.
"I met him yesterday for the first time and it was a big deal for me; it's been a special past 24 hours.
"We talked how I prepare, how much I play, where I go next; we exchanged ideas about our respective sports. That's what you talk about.
"He was my all-sports idol growing up. [NBA] was big on the German TV station, the NBA highlights. That's where I saw him do all his moves. I was into his great athleticism. That's what inspired me."