British number one Robson destroyed Poland’s world number four, the highest-ranked player the world number 41 has beaten in her career so far.
Robson, 19, has beaten some big names in the past, most notably former world number one Kim Clijsters, 2011 Wimbledon champion Petra Kvitova and Chinese star Li Na, but none of those three were in the top five at the time.
It is a huge boost for Robson, who has struggled so far this season after a breakthrough 2012. Only last week she suffered her sixth first-round defeat of the year at the Portugal Open.
Last year's Wimbledon finalist Radwanska has a reputation for being one of the more intelligent and consistent players on the women's tour, but she is relatively slight and has been known to struggle against big hitters like Robson.
Two such powerful returns helped the Briton claim the first break of the match as she moved to 5-3 in the first set, before going on to dominate the second following an exchange of early breaks, completing the win in one hour and 14 minutes.
Radwanska was the biggest casualty as second seed Maria Sharapova of Russia eased past Romanian qualifier Alexandra Dulgheru 7-5 6-2 and third-seeded Belarussian Victoria Azarenka had to fight past Russian Anastasia Pavlyuchenkova 7-6 7-6.
Suarez Navarro, who has never won a WTA tour title, was playing her first match since losing to Pavlyuchenkova in the final in Oeiras on Sunday.
"We only had a small window to prepare for today's match, but I was really focused," she said on the WTA website.
"Once I got here to Madrid, I didn't think of what happened in Oeiras for a single moment," the 24-year-old added.
"I just wanted to play the best I could here today. We only have one tournament here in Spain so for us it's a really important week."
World number one and top seed Serena Williams continues her Madrid title defence with a second-round match against Spaniard Lourdes Dominguez Lino on Tuesday.
Next up for Robson is the winner of the second-round match between 16th seed Ana Ivanovic and qualifier Chanelle Scheepers.
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- Kim Clijsters
- Petra Kvitova