The French teenager, ranked 87 in the world, won 117 points to the German's 116, but held her nerve in the decisive tie-break to come through to face the winner of Yanina Wickmayer's clash with Anastasia Pavlyuchenkova.
It could have been over sooner, with the 19-year-old taking the first set 7-5 and in control of the second. But Goerges dug deep to force a decider.
It was a good day for French hopes as Alize Cornet also emerged from a tricky match in front of her home fans to beat Sofia Arvidsson.
World number 35 Cornet raced into a 5-0 lead in the opening set, but Arvidsson fought back to make it a contest before succumbing.
Arvidsson, ranked just two places below her French opponent, then kept the momentum as she cruised through the second set to force a decider.
But Cornet then upped her game once more to reel off six consecutive games to make the last-16 and make up for her first-round exit in the event last year.
Elsewhere Roberta Vinci, seeded fifth in Paris, needed just 52 minutes to romp past Simona Halep 6-0 6-3 in the evening match.
Round 1 matches on Tuesday:
Alize Cornet (France) beat Sofia Arvidsson (Sweden) 6-3 4-6 6-0
Carla Suarez Navarro (Spain) beat Monica Niculescu (Romania) 6-3 7-6(3)
Stefanie Voegele (Switzerland) beat Tsvetana Pironkova (Bulgaria) 6-4 6-2
Kristina Mladenovic (France) beat 7-Julia Goerges (Germany) 7-5 5-7 7-6(4)
5-Roberta Vinci (Italy) beat Simona Halep (Romania) 6-0 6-3