The 23rd-seeded Vekic is playing in the second week of a Grand Slam for the first time and secured her first win against Goerges who has beaten her on three previous occasions.
“I don’t even know how I won this match,” said Vekic in an on-court interview. “I was just trying to get returns in ... I just kept fighting and believing I could win.”
Next up for the 23-year-old Croatian is Swiss Belinda Bencic, who earlier on Monday upset world number one and defending champion Naomi Osaka of Japan.
Number 26 seed Georges' strong serving took her into an early 4-1 lead in the first set but Vekic broke back to get to a tiebreak, the first played between the pair.
At 5-5, Georges hit a forehand winner down the line and took the set on serve with another forehand into the open court.
Goerges broke Vekic early in the second and had the chance to serve out the match at 5-4 but nerves got the better of her.
She served three double faults, failed to convert a match point and earned a time violation for serving too slowly as Vekic broke back and went on to level the match.
Goerges struggled to hold serve early in the third, eventually being broken in the fourth game. Vekic held on to her lead and served for the match at 5-4.
A pair of strong returns, a forehand winner and a stab volley put Goerges back within striking distance at 15-40.
But that lead was cancelled out by a string of forehand errors, enabling Vekic to get back to deuce. The Croat then followed up with a forehand winner to reach match point.
Goerges hit two deep shots, putting Vekic on the defensive and forcing her to pop up a relatively short lob which Goerges hit long. The German challenged the call, but Hawkeye confirmed that the ball was out giving Vekic the victory.
The Croat last met Bencic, who is seeded 13th, at the French Open this year where she won easily 6-4, 6-1.