Williams showed the form that once delivered her seven Grand Slam titles.
Ivanovic couldn't take advantage of the two break-point opportunities Williams offered in the sixth game of the first set.
"I'm just trying to play well every day, take one point at a time and learn from my mistakes every match, try to go out there relaxed with a clear mind," said Williams, who won back-to-back Dubai titles in 2009 and 2010.
A wild card entrant, Williams lost only seven total points in her other six service games as she moved to a 9-2 record against Ivanovic. However, the Serbian had won their last encounter in a three-set Auckland final last month.
"We've played many times in the past," Ivanovic said. "I definitely think she was a lot more aggressive (than in Auckland). She was stepping in the court way more, coming to the net more often."
"The conditions for me are tough," Radwanska said. "The ball was really flying.
"When I don't feel the ball, it's not good."
She led 4-1 then lost eight games in a row to be 6-4, 3-0 down.
Radwanska, the 2012 Dubai champion, also was ousted in the second round at the Qatar Open in Doha last week.
In another upset, third-seeded and defending champion Petra Kvitova of the Czech Republic joined Radwanska on the sidelines.
Kvitova had a 6-1, 4-2 lead, but couldn't close the door on tenacious Spaniard Carla Suarez Navarro, who won their two hour, 16 minute battle 1-6, 6-4, 7-6 (4).
"I start really well, I play pretty aggressively, I made a lot of winners, and she didn't have really time to hit well as she did after," Kvitova said. "And when I was leading in the second set, I lost very badly my serve, and I think that from that moment I was like a little bit mentally down."
Wozniacki defeated Annika Beck of Germany 6-4, 6-4.