Starting his second-round match at around 2100 local time on Rod Laver Arena, the six-time champion appeared to be in a hurry to finish the contest, breaking his opponent's serve twice in the first two sets and three times in the third.
Krajinovic took a medical timeout to have his right upper arm attended with Federer leading 2-1 in the third set but it made no difference to the momentum for the 20-times Grand Slam champion, who hit his 42nd winner to end the contest in a little over 90 minutes.
He said: "I'm very happy, it's a great start to the season, I'm feeling really relaxed out on court. You train hard and you hope it pays off."
Federer next faces Australian John Millman, who he lost to at the US Open in 2018.
Reporting by Reuters.