Jabeur, last year’s runner-up, won 5-7 7-6 (7-5) 6-3 on Arthur Ashe Stadium against Czech 31st seed Bouzkova, who was hampered by a thigh injury.
Tunisia’s Jabeur, 29, has been struggling with illness in New York.
“Hopefully I’ll continue with this warrior symbol,” she said after winning in two hours and 56 minutes.
“I love fighting. I trust my body. I know I can go all the way playing three sets, no problem.
“It’s not fun starting being sick. I feel like this tournament is testing me and showing me that I have a lot of strength, that I can go really far.”
Jabeur’s next match at Flushing Meadows as she hunts a first major title will be against Chinese 23rd seed Zheng Qinwen.
In the night session on Louis Armstrong Stadium, Wimbledon champion Marketa Vondrousova needed only 56 minutes to beat Ekaterina Alexandrova.
Czech ninth seed Vondrousova quickly wrapped up a 6-2 6-1 win against the Russian 22nd seed in a match which started at about 23:30 local time.
She goes on to play American world number 59 Peyton Stearns, who ended British number one Katie Boulter’s run.
Belarusian second seed Aryna Sabalenka also breezed through with a 6-1 6-1 victory over France’s Clara Burel.
Since winning the Australian Open earlier this year, Sabalenka has enjoyed runs to the semi-finals of the French Open and Wimbledon.
If she beats the result of defending champion Iga Swiatek in New York then she will replace the Pole as world number one.
Sabalenka has cruised through her opening-round matches without dropping a set and will face 13th seed Daria Kasatkina next after the Russian beat Belgium’s Greet Minnen 6-3 6-4.
Third seed Jessica Pegula fought off a comeback from Ukraine’s Elina Svitolina, maintaining her bid to become the first home singles champion since Sloane Stephens in 2017.
The 29-year-old American has reached the quarter-finals of six Grand Slam tournaments without going beyond the last eight.
Pegula will face compatriot Madison Keys in the fourth round after a 6-4 4-6 6-2 win against 26th seed Svitolina.
Keys, who lost to Stephens in the 2017 final, came from behind to win 5-7 6-2 6-2 victory against Russia’s Liudmila Samsonova.
