Kuznetsova, the 20th seed of Russia, won a lead-up event in Washington DC, and beat top-10 player Eugenie Bouchard in a promising preparation for the major at Flushing Meadows - which she won in 2004.
But there would be no happy 10th anniversary of her maiden major triumph in New York, Kuznetsova edged by New Zealand's Marina Erakovic 3-6 6-2 7-6 (7-3) in two hours 19 minutes.
The win was Erakovic's first at the American major, after four previous attempts in the main draw.
Slovakia's Cibulkova, meanwhile, suffered a stunning defeat at the hands of 15-year-old wildcard Catherine Bellis.
Bellis, who earned her main-draw entry by winning the US under-18s national championships, shot out of the blocks and fended off a comeback from the world number 13 to post a momentous victory 6-1 4-6 6-4.
The majority of seeds progressed comfortably, though, including American Serena Williams, third seed Petra Kvitova and eighth seed Ana Ivanovic.
Williams, the two-time defending champion and top seed, was a convincing 6-3 6-1 winner over compatriot and wildcard Taylor Townsend.
Kvitova, the two-time Wimbledon champion, obliterated France's Kristina Mladenovic, the Czech progressing in 54 minutes 6-1 6-0.
Serbia's Ivanovic was a 6-3 6-0 victor over American Alison Riske.
Canadian seventh seed Bouchard had a fitness question mark on her heading into the year's final major, but was able to see off Belarus' Olga Govortsova with minimal fuss 6-2 6-1.
American 27th seed Madison Keys dismantled Australian wildcard Jarmila Gajdosova 6-0 6-3, while 2011 champion Sam Stosur got through - the 24th seed downing American Lauren Davis 6-1 6-4.
Russian Ekaterina Makarova eased past American Grace Min 6-4 6-2, while Victoria Azarenka - the Belarusian 16th seed - endured an almighty challenge from Japan's Misaki Doi before advancing 6-7 (3-7) 6-4 6-1.
Italian 11th seed Flavia Pennetta got past German opponent Julia Goerges 6-3 4-6 6-1, while Goerges' compatriot Mona Barthel ousted Chinese 32nd seed Zhang Shuai 6-1 6-2.
Czech 30th seed Barbora Zahlavova Strycova was too good for Australia's Ash Barty 6-1 6-3, while Spain's Carla Suarez Navarro - the 15th seed - downed Barty's newfound compatriot Ajla Tomljanovic 3-6 6-2 6-1.
Russian Anastasia Pavlyuchenkova, seeded 23, thrashed Brazil's Teliana Pereira 6-2 6-0, while Australian 29th seed Casey Dellacqua pulled through with a 7-5 6-3 result over Austria's Patricia Mayr-Achleitner.
American Nicole Gibbs had a solid win over France's Caroline Garcia 6-2 2-6 6-3, while Vania King also progressed for the locals against major winner Francesca Schiavone of Italy 6-3 3-6 6-3.
Romania's Sorana Cirstea found some form against Great Britain's Heather Watson, winning 6-1 6-1, while American Coco Vandeweghe came from behind to beat Croatia's Donna Vekic 2-6 6-3 6-1.
Christina McHale continued the winning ways for the Americans, edging South Africa's Chanelle Scheepers 6-2 1-6 7-6 (7-5), while Varvara Lepchenko also entered the second round courtesy of a 7-5 6-2 win over Belgian Alison Van Uytvanck.
Shelby Rogers was another local winner, downing Maryna Zanevska of Ukraine 6-4 6-3, while China's Wang Qiang thrashed Poland's Paula Kania 6-2 6-0.
Petra Cetkovska won an all-Czech battle over Klara Koukalova 6-1 2-6 6-3, the latter's record at the US Open now reading 0-11.
Russian Elena Vesnina saw off Taipei's Chan Yung-Jan 6-0 7-5, Romania's Monica Niculsecu fought back to beat Kazakh Yaroslava Shvedova 6-7 (5-7) 6-1 7-5 and Estonian Kaia Kanepi got the better of France's Pauline Parmentier 7-6 (7-3) 3-6 6-1.
Ukrainian pair Lesia Tsurenko and Elina Svitolina both exited, Tsurenko losing to Kazakhstan's Zarina Diyas 6-1 6-2 while Svitolina was humbled 6-2 7-6 (7-4) by Slovenia's Polona Hercog.
Serbia's Aleksandra Krunic was too good for Poland's Katarzyna Piter while Czech Karolina Pliskova defeated Austria's Yvonne Meusburger.