The home favourite looked to be in control of the match at the WTA International clay-court event as she took the opening set, only for Scheepers to rally and record a 4-6 6-1 6-2 success.
Meusburger cut a frustrated figure on court as she struggled with her serve throughout the match, the world number 38 only held two service games as Scheepers converted 10 of the 21 break opportunities to progress.
The South African's reward is a last-eight match with second seed Sara Errani after the Italian completed a comfortable straight-sets victory over Paula Ormaechea.
Errani served well throughout the match with 83 per cent of her first serves landing in as she won 61 per cent of points to give her opponent little chance.
The world number 14 rattled off 11 consecutive games to secure victory as she put a nervy start behind her to progress 6-3 6-0.
There was another surprise earlier in the day as third seed Carla Suarez-Navarro, like Meusburger, exited earlier than expected.
Having battled valiantly in the opening set, the Spaniard's game deserted her and Shelby Rogers, her 21-year-old opponent, took full advantage to take a 6-4 6-0 win.
Rogers will meet another seed in the quarter-finals after Camila Giorgi, the seventh seed, beat Lisa-Maria Moser 6-0 6-2.