The Belarusian has been trying to turn around a dismal second half of the season, having failed to reach a final since winning at Indian Wells and the Miami Open to complete the Sunshine Double in March.
She seemed well on her way to flipping the script with the crowd on her side at Arthur Ashe Stadium.
Sabalenka, who would join Serena Williams and Chris Evert as the only women to win three straight US Opens in the professional era, fired off 32 winners to set up a meeting with Russian Polina Iatcenko in the second round.
"I'm super happy with the level I played," said Sabalenka. "I feel like this is one of my favourite stadiums, I really experienced the best emotions here on this stadium."
Osorio handed Sabalenka a break with a double fault in the third game - one of five she had across the opening set - and slipped 4-1 behind as the Belarusian converted on another break point with an unreturnable backhand.
The Colombian trimmed the lead to 4-2 in the next game but never got a hold of the momentum and her racket went flying in her futile attempt to reach Sabalenka's perfectly placed drop shot on break point in the seventh game.
After trading early breaks in the second set, Sabalenka wore down her opponent's defence to secure the pivotal break in the ninth game, before closing it out with an unreturnable serve and a subdued celebration.
Sabalenka is also defending her 99-week long world No.1 ranking in New York, with three other players in the women's main draw in a position to overtake her.
Amanda Anisimova, the player beaten in last year's final, breezed past Ashlyn Krueger 6-1 6-3.
Playing on her 25th birthday, Anisimova fought off three break points early for a 2-1 lead before running away with the opening set in the all-American match-up at Louis Armstrong Stadium. The world No. 10 wrapped up the win in 72 minutes.
After the match the crowd sang "Happy Birthday" to Anisimova who will face Austrian Lilli Tagger in the second round.
Elsewhere, seventh seed Karolina Muchova, the Wimbledon runner-up, thrashed Sinja Kraus 6-3 6-0 and will now play Alex de Miinaur's wife, Briton Katie Boulter.
Another Briton, Harriet Dart, beat home hope Peyton Stearns 3-6 6-3 6-2 clinching victory on her fourth match point and next plays 25th-seed Marie Bouzkova who beat Elsa Jacquemot 6-2 6-2.
The Olympic champion Zheng Qinwen, who was forced to go through qualifying two years after reaching the quarter-finals in New York, beat Kristina Liutova 4-6 6-3 6-1.
Spain's Cristina Busca put out 28th-seed Sara Bejlek 7-5 4-6 6-1 but Emma Navarro, seeded 26 both progressed.
with agencies
