After dominating the first set in Stuttgart on Thursday, fifth seeded French Open and Wimbledon finalist Paolini also opened a 2-0 lead in the second.
Wildcard Niemeier fought back to 2-2, and survived another break in the seventh game en route to a 5-4 lead.
But Paolini broke through again for 6-5 and served out the match in the next game after one hour, 26 minutes.
Also into the last eight is Ekaterina Alexandrova, who ousted sixth seeded Mirra Andreeva, 6-3 6-2.
Meanwhile, third seed and last year's US Open finalist Jessica Pegula made it to the quarters with a 6-1 6-1 victory over Magdalena Frech of Poland.
Also into the last eight is Ekaterina Alexandrova, who ousted sixth seeded Mirra Andreeva, 6-3 6-2.
Set to play later was American fourth seed Coco Gauff, who will face Germany's Ella Seidel.
Because of the Good Friday holiday, the quarter-finals are played on Saturday, and the indoor clay tournament runs until Monday.
Also in the quarters are world number one Aryna Sabalenka of Belarus and second-ranked Iga Swiatek of Poland.
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