The Polish crowd favourite moved two games clear only to see veteran Meusburger restore parity at 2-2 in the first set.
But Radwanska would win eight of the next nine games from 3-3 to progress with some ease.
"The court was a bit different from that in the first two matches," said Radwanska.
"It took some time before I got used to it, but the most important thing is that I got through."
It was a 21st win of the year for Radwanska as she tied Li Na for the most wins of the year.
"One of the things I like most about this tournament is that it's not far from the place where I live, it's like a 45-minute drive away," said Radwanska.
"I got really excited when I noticed my beloved grandma Wieslawa beside my mother in the stands.
"She has always been one of my biggest fans and supporters, she has watched all my matches on TV or internet since the very beginning of my career. But never before had she seen me play live, and now she can come to see all of my matches."
"Her presence gives me positive vibes. I'm so happy!"
Radwanska will face Alize Cornet in the semi-finals after she completed a 6-2 0-6 6-2 win over Klara Koukalova.
3-Carla Suarez Navarro (Spain) beat 7-Magdalena Rybarikova (Slovakia) 6-2 6-3
Camila Giorgi (Italy) beat Shahar Peer (Israel) 6-1 6-4
1-Agnieszka Radwanska (Poland) beat 6-Yvonne Meusburger (Austria) 6-4 6-1
4-Alize Cornet (France) beat 5-Klara Koukalova (Czech Republic) 6-2 0-6 6-2
- Sports & Recreation