Jabeur, in her first Grand Slam since suffering "the most painful loss" of her career in the Wimbledon final and a day after turning 29, showed a warrior's mentality in a two-hour battle at Louis Armstrong Stadium played in humid conditions.
"It wasn't an easy match, she plays unbelievable and I'm not feeling my best today," Jabeur said in her on-court interview before the crowd sang 'Happy Birthday' to her.
"I know at some point I didn't have the best attitude on the court but I was trying, you know, to get to play.
"It's not easy also playing her. She was trying to make me run even more so that wasn't helping. But I'm glad that I got the win, especially that I showed myself that I can push and do better on the court."
Jabeur, who lost the last two Wimbledon finals and suffered the same fate at the 2022 US Open, was in total control early as she went up a double break for a comfortable 4-1 lead before her energy level and serve percentage started to drop.
While serving with a 4-3 lead Jabeur told the chair umpire she was having trouble breathing and then went on to lose the game as Osorio broke to get the match back on serve.
When Osorio won a fourth consecutive game for a 5-4 lead Jabeur looked out of sorts while she spent several minutes in her chair as a doctor took her blood pressure and did a general health check before the Tunisian resumed the match.
A determined Jabeur dug deep and won three consecutive games to wrap up the opening frame in 59 minutes before leaving the Louis Armstrong Stadium court.
Both players struggled to defend their serve in the second set where Jabeur broke for a 4-3 lead and, with the finish line in sight, turned aside two break points to hold for a 5-3 advantage.
But Osorio refused to back down and saved two match points on her next service game and then broke to love to level the set at 5-5 before another exchange of breaks forced a tiebreaker where Jabeur found her way on her third match point when the Colombian sent a forehand into the net.
Up next for Jabeur will be unseeded Czech Linda Noskova.