The Pole trailed by a set and a break to German Niemeier, who showed her potential by reaching the quarter-finals of Wimbledon in just her second main draw appearance at a grand slam.
Swiatek has not been in great form since her 37-match unbeaten run was ended by Alize Cornet at Wimbledon, and she made plenty of errors again here.
But the world number one fought hard to gain the ascendancy in the second set and dominated the third to win 2-6 6-4 6-0.
Swiatek said: "I'm just proud that I didn't lose hope because she was playing really good in the first set. I'm pretty glad that I used my experience to keep the same kind of level throughout the whole match."
In the quarter-finals Swiatek will face American Jessica Pegula, who is through to the last eight at a slam for the third time this season following a 6-3 6-2 victory over Petra Kvitova.
Pegula is part of a famous American sporting family, with her parents Terry and Kim Pegula the owners of NFL team the Buffalo Bills and NHL team the Buffalo Sabres.
She is enjoying the extra support that brings, saying: "It's definitely different playing overseas, I feel people don't make the connection as often as they do here.
"There is a lot of people from Buffalo that are kind of all over the place, you always find somebody that's a Bills and Sabres fan.
"And to me I think I have more embraced it and find it fun and cool that these people, because there's not a lot of tennis in Buffalo, know nothing about tennis but they will come and watch the match in Cincinnati or here or wherever, just because they are a fan of the teams."