Beck had a total of 11 game points during the match - three in the third game of the first set, four in the first game of the second set and four more in the fourth game of the second set - but the No.2-seeded Radwanska wouldn't let her have any of them, winning all 12 games in an hour and seven minutes.
It wasn't Radwanska's first 60 60 win at the WTA main draw level - it was her third, having also done it at Istanbul in 2008 (against Sorana Cirstea) and in the final of Sydney in 2013 (against Dominika Cibulkova). She has done it once in WTA qualies and once at the ITF Women's Circuit level, as well.
Other winners included No.10 seed Caroline Wozniacki, who battled over two and a half hours to beat Kazakhstani qualifier Yaroslava Shvedova, 76(7) 36 61, Australian qualifier Casey Dellacqua, who beat No.13 seed Roberta Vinci, 46 64 63, and Lauren Davis, who beat Varvara Lepchenko, 63 62.
Wozniacki had lost her only previous meeting against Shvedova - on the clay of Madrid last year.
"Playing her is always tricky, especially here because the ball bounces so high - her serves were bouncing way over my head and I had a really tough time getting into the rhythm at the start of the match," Wozniacki said. "I managed to fight through and win that first set. I'm not sure how, I guess my running really got me through that. And then I started playing some pretty good tennis in the third.
"But before the third set it was really just a battle and a scramble out there today."
Was it was a revenge mission? "Our first match was on clay and she played very well - I didn't play well at all in that match," Wozniacki said. "You know, it's a new tournament, a new place. I just look at matches she's played this year, because it's more recent, then just try and see a few key points."
Jankovic and Wozniacki, both Grand Slam finalists and former World No.1s, will now square off in an all-star clash in the fourth round. Wozniacki leads Jankovic in their head-to-head series, 5-4, losing their first four but winning their last five - they haven't played in almost two years, however.