The 20-year-old American, who was runner-up at last month’s Family Circle Cup in her homeland, lost the first set 6-4 against her Croatian opponent.
But Keys, looking for her second career title in France, narrowly moved past Lucic-Baroni with scores of 7-6 (7-3) 7-6 (7-0) in the second and third sets to reach the last eight.
Home favourite Alize Cornet, who won here in 2013, was not so lucky as she was knocked out after taking the first set against Russia’s Elena Vesnina.
The 25-year-old fourth seed suffered a 3-6 6-0 6-3 loss to Vesnina in an hour and 40 minutes in Strasbourg.
Also on Wednesday, another Frenchwoman in Kristina Mladenovic made it through with a 7-6 (7-4) 6-2 win over America’s Alison Riske, while Italian wildcard Francesca Schiavone lost 2-6 3-6 against France’s Virginie Razzano.