The Australian, seeded seven this year at Flushing Meadows, was far too strong for her Croatian opponent, winning the first 19 points of the match in a row, then going on to hit 10 aces and landing 69 per cent of her first serves.
"I think probably today I carried on from exactly how I was feeling last year," said Stosur, who upset Serena Williams in last year's final.
"Hopefully that's a good omen. As far as first rounds go, I was very happy with today. Last year I did feel that comfort out there and nothing bothered me; today was along the same lines."
Stosur lost in the first round of the Australian Open and the Olympics and was beaten in round two at Wimbledon.
But the 28-year-old showed no signs of nerves and now plays Edina Gallovits-Hall of Romania after she beat Stefanie Voegele of Switzerland 7-5 6-4.
The match was over in just 51 minutes on Arthur Ashe Stadium - which was quick enough to beat the deluge that then descended over New York.
Just a few minutes after the match, play was suspended on all courts with a lengthy rain delay expected.
Another former champion Maria Sharapova cruised into the second round with a 6-2 6-2 victory over Hungary's Melinda Czink.
The Russian, who won the title in 2006 and is seeded third this year, coped well with a swirling wind to see off the world number 88 in 67 minutes.
Sharapova broke Czink's serve twice to win the first set and reeled off the last three games to set up a clash with Spain's Lourdes Dominguez Lino, who beat Sesil Karatantcheva (Kazakhstan) 6-4 6-1.
The 25-year-old is chasing her second Grand Slam title of the year, having won the French Open in June to complete a career set of all four Grand Slam events.
World number one Victoria Azarenka wasted little time in outlining her title credentials as she scorched to a 6-0 6-1 first round victory over Russia's Alexandra Panova.
The Belarusian, who won her first Grand Slam title at the Australian Open in January, was at her very best as she raced through in just 50 minutes.
Azarenka has yet to go beyond the last 16 at the US Open but she outclassed world number 76 Panova with a performance full of power and precision.
The 23-year-old dropped just five points in the first set and maintained her momentum in the second to cruise into round two.
Kim Clijsters overwhelmed American teenager Victoria Duval 6-3 6-1 in a powerful start to the final tournament of her career.
The three-times champion at Flushing Meadows wobbled a bit in the opening set against the 16-year-old wildcard but prevailed in 51 minutes.
Duval was the youngest player in the women's draw and won three straight games at one point in the first set against the former world number one.
Clijsters, 29, will retire after the US Open to concentrate on family life.
In other matches completed on the first day of action at Flushing Meadows, Petra Kvitova, seeded fifth, saw off Slovenia's Polona Hercog 7-6(6) 6-1, while number nine seed Li Na defeated Britain's Heather Watson with a comfortable 6-2 6-3 win.
France's 11th seed Marion Bartoli was made to work by American Jamie Hampton, before winning through 6-3 7-6(5) and there were no problems for Lucie Safarova, seeded 15, who beat another US player Melanie Oudin 6-4 6-0.
Russian duo Anastasia Pavlyuchenkova and Nadia Petrova, seeded 17 and 19 respectively, booked their places in the second round with wins over Daniela Hantuchova of Slovakia and Australian Jarmila Gajdosova, while Belgium's 25th seed Yanina Wickmayer beat Israel's Julia Glushko to ensure her progress.
Two Czech players produced upsets as Kristyna Pliskova knocked out 18th seed Julia Goerges and Lucie Hradecka downed 27th seed Anabel Medina Garrigues.
And it was a day to forget for 16th seed Sabine Lisicki, who fell 4-6 6-2 6-2 to Sorana Cirstea of Romania.
- Flushing Meadows