Radwanska battled past her opponent by sealing a 7-6(1) 6-3 victory in one hour and 38 minutes in Singapore.
The first set was a tight affair as both competitors broke serve on three occasions, and it was Muguruza who forced the set into a tie-break after saving set point.
The 23-year-old in my was unable to keep her momentum, though, as Radwanska broke serve three times to comfortably win the breaker 7-1.
It took an hour to get through the first set, but just 38 minutes for the second as Radwanska recovered from losing serve by winning three games in a row after taking it to 3-3.
Radwanska will play Karolina Pliskova for a place in the semi-finals