Schiavone was 25 of 30 on first serve points in her 6-3, 7-6(4) first round defeat of Alexandra Dulgheru, improving to 3-0 against the Romanian without having dropped a set.
It also was Schiavone's 554th professional win, tying her with Flavia Pennetta for sixth-most among active players.
As for the second-seeded Garcia, after dropping the first set against Timea Babos in a tie-break, the Frenchwoman jumped out to a 4-0 second set lead and went on to a 6-7(4), 6-1, 6-3 triumph.
"I was not relaxed at the beginning of the match, but I tried to keep calm and play my game, and it got better," Garcia said. "I'm very happy to have won this first match."
Four quarterfinal spots were also secured, two of them by teenagers.
The youngest player in the draw, 16-year-old Ana Konjuh, was a 6-3, 6-1 winner against fellow teen Carina Witthoeft. Next up for the Croat is 18-year-old French wildcard Océane Dodin, who won 11 straight games from 4-1 down in the first set against qualifier Katarzyna Piter for a 6-4, 6-0 victory. In fact, she was a point away from a 5-1 first set deficit but fired off three straight aces to start her run.