At WTA Memphis, British number one Heather Watson - seeded fourth for the tournament - battled through a topsy turvy match against Andrea Hlavackova, winning 2-6 6-0 6-3.
In Dubai seven of the world's top 12 remain in the draw, but the tournament has been shorn of its two top names, with Australian Open champion Victoria Azarenka pulling out on Tuesday because of a foot injury.
World number one Williams said she had been struggling for several weeks with the back injury and is targeting a return at next month's Miami Open after pulling out of her Dubai second-round match just before she was due to play.
Earlier, world number four Radwanska beat teenager Putintseva 7-5 6-3 despite losing three of her first four service games. The Pole had a bye into the second round.
"For me, the first match is always a bit tricky," said Radwanska, 23.
"I didn't hit on the centre court here this year, and I was just practising outside - it was actually much faster. I was surprised - the ball is just not going anywhere."
Toiling at 4-3 down, Radwanska came to the net more often to force Putintseva, ranked 101 in the world, on to the back foot.
The Kazakh, who beat Britain's Laura Robson in an all-teenage first-round battle, prefers brute force over finesse and although a fast hitter, she consistently played the ball too short, allowing Radwanska to stay in rallies with relative ease.
Radwanska, the top remaining seed, was too out of sorts to fully exploit her opponent's shortcomings but as the overcast sky gave way to a muggy desert sun she also upped the playing temperature on court and took control of the match.
She will have to improve to defeat her next opponent, Czech Petra Kvitova, who beat former world number one Ana Ivanovic 7-5 7-6.
The 24-year-old-Serb started with a 4-1 winning record against the 2011 Wimbledon champion, but three of those victories came in 2008-9 when Ivanovic appeared poised to become the new force in the women's game.
She has sunk down the rankings since, although showed enough of her undoubted talent to trouble the world number eight in Dubai.
Ivanovic battled back from 5-1 down in the first set to square the match at 5-5, saving three set points along the way, but just when she appeared in the ascendancy, Kvitova broke again and eventually converted a fourth set point.
Ivanovic sparkled in patches and was defter in her touch, but ultimately could not cope with the relentless power of the 22-year-old's brutal baseline play.
"She's a clean striker of the ball, it's hard to pick where she's going to go - you sort of have to commit to one side," said Ivanovic. "She served big when she needed it."
Another former world number one, Caroline Wozniacki, routed China's Jie Zheng 6-0 6-1 to earn a quarter-final showdown with France's Marion Bartoli, who had been due to play Williams.
Samantha Stosur - a 6-4 6-0 winner against Taiwan's Su-Wei Hsieh - will play Italy's Roberta Vinci, who ousted German Angelique Kerber 7-5 6-1.
WTA Dubai results:
5-Sara Errani (Italy) beat Sorana Cirstea (Romania) 6-4 6-4
Marion Bartoli (France) beat 2-Serena Williams (U.S.)WO
6-Petra Kvitova (Czech Republic) beat Ana Ivanovic (Serbia) 7-5 7-6(1)
Roberta Vinci (Italy) beat 4-Angelique Kerber (Germany) 7-5 6-1
7-Samantha Stosur (Australia) beat Hsieh Su-Wei (Taiwan) 6-4 6-0
8-Caroline Wozniacki (Denmark) beat Zheng Jie (China) 6-0 6-1
3-Agnieszka Radwanska (Poland) beat Yulia Putintseva (Kazakhstan) 7-5 6-3
Nadia Petrova (Russia) beat Carla Suarez Navarro (Spain) 6-1 6-0
WTA Bogota results:
Teliana Pereira (Brazil) beat 2-Alize Cornet (France) 7-5 6-7(2) 6-2
Maria Teresa Torro (Spain) beat Mathilde Johansson (France) 7-6(3) 6-3
Mandy Minella (Luxembourg) beat 8-Timea Babos (Hungary) 6-1 6-4
Mariana Duque (Colombia) beat Estrella Cabeza Candela (Spain) 6-3 3-6 6-4
WTA Memphis results:
8-Kristina Mladenovic (France) beat Victoria Duval (U.S.) 6-1 3-6 6-3
3-Sabine Lisicki (Germany) beat Melinda Czink (Hungary) 7-6(0) 4-6 6-2
Marina Erakovic (New Zealand) beat 2-Sofia Arvidsson (Sweden) 6-1 3-6 6-3
4-Heather Watson (Britain) beat Andrea Hlavackova (Czech Republic) 2-6 6-0 6-3
1-Kirsten Flipkens (Belgium) beat Lesia Tsurenko (Ukraine) 7-5 3-6 6-3
Stefanie Voegele (Switzerland) beat Claire Feuerstein (France) 3-6 7-6(2) 6-1