Serena, despite being seeded only fourth at Flushing Meadows, is in no doubt she is the world's best player.
"Of course I believe that," said Serena after powering to a 6-1 6-1 win over Coco Vandeweghe after less than an hour on court. "I think there's a number of players on this tour, a few players who believe that.
"I don't care about the ranking. I've been number one. It's cool. But my thing is just to be the best player. If that means I'm winning and I'm not number one, that's fine."
Serena said it was difficult to gauge her form because the strong wind inside Arthur Ashe Stadium made it difficult to find her rhythm.
"I usually gauge the wind by my hair," she said. "If it's really going forward I'm thinking, it's really windy.
"I mean, the match was so weird, the conditions were so tough, I couldn't really play my game. It's not the best opportunity to grade yourself."
Serena will meet Spain's Maria Jose Martinez Sanchez in the second round.
Her one-sided victory came after sister Venus defeated Bethanie Mattek-Sands in straight sets, leading the older Williams to predict a return to the top 10.
Venus landed a first serve of 124 mph during her 6-3 6-1 demolition of her fellow American to set up a re-match with the woman who beat her in the third round of the Olympics, Germany's sixth seed Angelique Kerber, who eased past Britain's Anne Keothavong 6-2 6-0.
The former world number one has fallen to 46th on the rankings but claimed she had the weaponry to return to the elite.
"I'm looking forward to the top 10, all that great stuff," the 32-year-old Venus said. "I feel like I have it in me."
The seven-times major champion produced a vintage serving display against Mattek-Sands and said the importance placed on her delivery occasionally came at the detriment of the rest of her game.
Williams won 83 per cent of the points on her first serve and clubbed 22 winners in a powerhouse performance.
"My serve is huge - it's such a huge part of my game," she said. "When there's a big point, I'm thinking my serve is going to help me every time. I get so involved in my serve I forget to play the rest of the point.
"When your serve is on, the rest of your game is beautiful."
Former world number one Caroline Wozniacki, hampered by a knee problem, was a first round casualty on day two of action in New York.
The eighth-seeded Wozniacki never broke the serve of world number 96 Irina-Camelia Begu of Romania and lost 6-2 6-2.
"You always want to go in and do your best no matter what's happening out there," Wozniacki said. "I tried. I didn't play particularly well, made too many errors.
"It's unfortunate because it's a huge tournament, a tournament you want to play well in."
Second seed Agnieszka Radwanska ignored a niggling shoulder problem to breeze past Russian world number 91 Nina Bratchikova.
The Wimbledon runner-up, who withdrew from the warm-up event in New Haven because of pain in her right shoulder attributed to a long season, enjoyed a speedy 6-1 6-1 win over Bratchikova in humid conditions at Flushing Meadows.
But the only sign of a shoulder problem on Tuesday was the strapping visible beneath her shirt and the 23-year-old Pole said it was something she was coping with.
"It's sometimes worse but it feels better right now," Radwanska said after her 54-minute opening match at the year's final Grand Slam. "That's about it.
"It's always different when I'm playing matches or I'm just practising and not pushing that much. Of course I had some rest before the US Open. I'm really taking care of it right now. It's really better."
Radwanska, who has never advanced beyond the last 16 at Flushing Meadows, had just six unforced errors as she advanced to a meeting with either Ksenia Pervak of Kazakhstan or Spain's Carla Suarez Navarro.
Radwanksa's sister Urszula did not have things go her way though as she was beating 6-1 6-1 by 20th seed Roberta Vinci of Italy.
Vinci's compatriot Sara Errani beat Garbine Muguruza Blanco of Spain 6-3 6-7(6) 6-1, while 13th seed Dominika Cibulkova came from behind to beat Johanna Larsson 6-7(5) 6-2 6-0.
Maria Kirilenko raced through her first-round match with Chanelle Scheepers, the Russian number 14 seed winning 6-2 6-1, while Jelena Jankovic saw off Kateryna Bondarenko in equally comprehensive fashion 6-1 6-2.
Number 12 seed Ana Ivanovic overcame a lack of hard court preparation and recent foot problems to cruise by 17-year-old Ukrainian qualifier Elina Svitolina, 6-3 6-2.and she will next play Sweden's Sofia Arvidsson who defeated Japanese veteran Kimiko Date Krumm of Japan 6-4 6-2.
Francesca Schiavone was stunned 6-3 6-4 by American Sloane Stephens, the Italian 22nd seed being joined on the scrapheap by 24th seed Klara Zakopalova who lost 6-4 6-4 to Andrea Hlavackova.
Other seeds to fall were Peng Shuai of China, who lost 7-6(5) 6-1 to Russia's Elena Vesnina and 26th seed Monica Niculescu of Romania, who was beaten 6-3 6-3 by Ayumi Morita.
Tamira Paszek, the 29th seeded Austrian, also tumbled out, 6-3 6-4 to Austria's Olga Govortsova and in the last match to finish on Tuesday, American 21st seed Christina Mchale fell to Dutchwoman Kiki Bertens 6-4 4-6 6-3.
- Flushing Meadows