Despite a lengthy rain delay midway through the match, Radwanska kept her composure to close it out, 63 63, and set up a last four meeting with Jamie Hampton.
"We all prepare for rain when we are playing somewhere like Auckland where the weather can change very quickly," Radwanska said about the two hour interruption at the start of the second set.
"I was so focused when I came back on court and I'm happy to get through because they were two tight sets. I knew that against her I would have to play very well from the first point to the last and fortunately I was able to play my best tennis," she added.
After racing through the opening set in just 26 minutes and then going up a break in the second, Hampton lost her way as Bertens fought back to level the match on a tie-break.
However, Hampton steadied the ship again in the third, capturing a couple of early breaks before closing out the match, 61 67(5) 62.
In a match dominated by serve, and interrupted by three rain delays, breaks towards the end of each set proved enough to see Wickmayer through, 63 64.
"It feels good to win another match and be through to the semifinals in the first week of the year," Wickmayer said. "It's nice to get some matches and get some confidence to bring to the Aussie Open.
"I had a good off-season, but matches are different to practice, so I'm pleased this week has gone so well and I've found my rhythm so fast."