The Australian second seed advanced to a second-round match against America's 73rd-ranked Lauren Davis.
"I probably would have preferred if it was a little bit more straightforward, but, yeah, happy the way we were able to fight through that and find a way after a pretty awful start," said Barty.
"I think I just didn't give myself a chance in that first set. Sort of appalling, probably made a set's worth of errors. I think it was just nice to kind of lock down a little bit. It took some time to kind of break her down in the end."
Barty is favorite to regain the number one ranking after the Flushing Meadows fortnight, with incumbent Naomi Osaka, Wimbledon champion Simona Halep and Kristina Pliskova also in contention.
Her opening match was Barty's first singles outing on the cavernous Arthur Ashe Stadium court, where she lifted the women's doubles title alongside CoCo Vandeweghe a year ago.
"It was cool. A bit of a unique feeling, bizarre feeling walking back out on to Arthur Ashe. It was really cool. Different to what it felt like during that match," said Barty.
"Obviously it was an incredible atmosphere, and then walking out to practice obviously has a bit of a different buzz.
"Incredible memories from my time here in New York last year, an extremely good fortnight of tennis, both in singles and doubles.
"I'm just glad we're off to a good start here and now we can start really getting our teeth stuck into the tournament."
The 23-year-old, who during a break from tennis played cricket for the Brisbane Heat in the Women's Big Bash League four years ago, has also been keeping a close eye on the ongoing Ashes series between England and Australia.
England levelled the five-match showdown with an incredible one-wicket win at Headingley on Sunday, as Ben Stokes produced one of the greatest innings in Test history with a remarkable 135 not out.
"Credit where credit's due. Stokesy was incredible. I think we missed some opportunities and my whole team were glued to it," said Barty, who returned to tennis in 2016 ranked 683 in the world.
"I have an English trainer, as well. He's had a fantastic 24 hours of feeding it to us and not letting us forget that result. Yeah, I mean, I'm glued to it. Another couple of Tests to go. We will enjoy those as well."
US Open results on Monday (x denotes seeding):
Novak Djokovic (SRB x1) bt Roberto Carballes Baena (ESP) 6-4, 6-1, 6-4
Denis Kudla (USA) bt Janko Tipsarevic (SRB) 3-6, 6-1, 7-6 (7/5), 6-1
Dusan Lajovic (SRB x27) bt Steve Darcis (BEL) 7-5, 6-3, 6-3
Jenson Brooksby (USA) bt Tomas Berdych (CZE) 6-1, 2-6, 6-4, 6-4
Nikoloz Basilashvili (GEO x17) bt Marton Fucsovics (HUN) 3-6, 6-4, 6-2, 3-6, 6-3
Daniil Medvedev (RUS x5) bt Prajnesh Gunneswaran (IND) 6-4, 6-1, 6-2
Dan Evans (GBR) bt Adrian Mannarino (FRA) 6-4, 6-3, 2-6, 6-3
Cristian Garin (CHI x31) bt Christopher Eubanks (USA) 3-6, 7-6 (7/5), 6-4, 6-7 (4/7), 6-3
Alex de Minaur (AUS) bt Pierre-Hugues Herbert (FRA) 6-4, 6-2, 6-7 (6/8), 7-5
Bradley Klahn (USA) bt Thiago Monteiro (BRA) 6-3, 6-2, 6-3
Kei Nishikori (JPN x7) bt Marco Trungelliti (ARG) 6-1, 4-1 retired
Venus Williams (USA) bt Zheng Saisai (CHN) 6-1, 6-0
Rebecca Peterson (SWE) bt Monica Puig (PUR) 6-3, 6-3
Dayana Yastremska (UKR x32) bt Monica Niculescu (ROM) 6-4, 1-6, 6-2
Johanna Konta (GBR x16) bt Daria Kasatkina (RUS) 6-1, 4-6, 6-2
Margarita Gasparyan (RUS) bt Priscilla Hon (AUS) 7-6 (7/4), 6-4
Ekaterina Alexandrova (RUS) bt Samantha Stosur (AUS) 6-1, 6-3
Ons Jabeur (TUN) bt Caroline Garcia (FRA x27) 7-6 (10/8), 6-2
Aliaksandra Sasnovich (BLR) bt Jennifer Brady (USA) 6-1, 4-6, 6-0
Mariam Bolkvadze (GEO) bt Bernarda Pera (USA) 6-3, 5-7, 6-4
Karolina Pliskova (CZE x3) bt Tereza Martincova (CZE) 7-6 (8/6), 7-6 (7/3)
Hsieh Su-Wei (TPE x29) bt Jana Cepelova (SVK) 6-4, 5-7, 6-3
Petra Martic (CRO x22) bt Tamara Zidansek (SLO) 6-4, 4-6, 6-1
Ana Bogdan (ROM) bt Harriet Dart (GBR) 6-3, 6-1
Iga Swiatek (POL) bt Ivana Jorovic (SRB) 6-0, 6-1
Anastasija Sevastova (LAT x12) bt Eugenie Bouchard (CAN) 6-3, 6-3
Kristina Mladenovic (FRA) bt Angelique Kerber (GER x14) 7-5, 0-6, 6-4
Fiona Ferro (FRA) bt Daria Gavrilova (AUS) 6-3, 6-4
Alison Van Uytvanck (BEL) bt Viktoria Kuzmova (SVK) 6-4, 6-4
Lauren Davis (USA) bt Johanna Larsson (SWE) 7-5, 6-2
Ashleigh Barty (AUS x2) bt Zarina Diyas (KAZ) 1-6, 6-3, 6-2