Jones broke serve early in her match and was 3-0 up after just 11 minutes at a windy stadium 17.
However, the 24-year-old had to battle to save seven break points before the end of the set as Gorgodze fought back from a poor start.
The Georgian cut a frustrated figure during the opening set – at one point she turned on spectators behind her who were asked by the umpire to switch off their mobile phones – but then began the second set with a break and raced into a 3-0 lead.
At times Jones struggled to cope with the 33-year-old’s powerful forehand, but the Briton broke for a third time to get the set back on serve.
Jones – the top seed in women’s qualifying – had to save four set points in a brutal tie-break before claiming victory with her fifth match point.
“If I could swear, I would!” Jones told Sky Sports after the match.
“It was really difficult, she [Gorgodze] makes you play a lot of balls. She played some really good tennis in the second set, but I just didn’t want to be out here in a third set today.”
Victory over Arianne Hartono in the third round of qualifying on Friday would secure Jones – who grew up playing on clay – a place in the main draw for the first time in her career.
She has never received direct entry to a Grand Slam through her ranking and narrowly missed out this time, despite having climbed to a career-high 86th in the world since the US Open cut-off point.
Fellow British player Harriet Dart is in qualifying action later on Thursday.
