After losing to Maria Sharapova in the second round of the U.S. Open, Babos struggled early and even trailed by a break point in the second set. In the third set, the Hungarian rolled past her American opponent.
Babos improved her hardcourt record to 12-13 and after beating Loeb, she is 8-4 against opponents ranked 101 or lower.
Babos will face Britian's Naomi Broady in the second round. Broady also needed three sets to get a 6-4, 3-6, 6-3 victory over Australian teenager Destanee Aiava.
Broady secured her win by recording 13 aces and losing serve only once.
In other opening round matches, Caroline Dolehide had little difficulty in beating Canadian Charlotte Robillard-Millette, 6-3, 6-2. Dolehide never faced a break point against Robillard-Millette, who converted just 16 percent of second-serve points and committed six double faults.
Dolehide will face the winner of the match between second-seeded Oceane Dodin of France and Canada's Gabriela Dabrowski.
New Zealand's Marina Erakovic also advanced to the second round with a 6-4, 6-1 victory over Russian qualifier Alla Kudryavtseva. Erkaovic will face seventh-seed Alison Van Uytvanck or wild card Carol Zhao.