Pennetta, who narrowly avoided being dumped out in the first round of qualifying, reeled off eight consecutive games to turn an ominous 4-1 deficit in the opening set into a commanding position.
Radwanska eventually brought the one-way traffic to a halt by breaking for 3-1 in the second set, but that proved the Pole’s final victorious game as Pennetta raced home for a 6-4 6-1 triumph.
"It was the first time I've beaten her in six years - I thought I would never beat her again," said Pennetta.
"But today I played really well. My idea of what I had to do was clear, and I tried to go for the points all the time.
"I made a few mistakes in the beginning - I was down 4-1, but every game was a big fight - but I kept going and stayed aggressive and it worked in the end."
Defeat looked inevitable for Navarro when the Spaniard trailed 6-1 4-2, but she responded magnificently to force a deciding set, eventually sealing the match 1-6 6-4 7-6(4).
World number 27 Cirstea beat Errani 6-2 5-7 6-1.
"Every single time against her I would come close but never get the win, but tonight I kept fighting and playing aggressive until the end," said Cirstea.
"I never changed my game plan, no matter what the score was. I'm happy because I kept doing that until the end, and that's what gave me the win."
Results from the Dubai Championships Women's Singles Round 2 matches
Alize Cornet (France) beat Kirsten Flipkens (Belgium) 6-3 6-4
Venus Williams (U.S.) beat Ana Ivanovic (Serbia) 6-2 6-1
8-Caroline Wozniacki (Denmark) beat Annika Beck (Germany) 6-4 6-4
Flavia Pennetta (Italy) beat 2-Agnieszka Radwanska (Poland) 6-4 6-1
Results from the Rio de Janeiro Open Women's Singles matches
1-Klara Zakopalova (Czech Republic) beat Silvia Soler Espinosa (Spain) 6-4 6-1
Nastassja Burnett (Italy) beat Alison Van Uytvanck (Belgium) 0-6 6-3 6-2
Katarzyna Piter (Poland) beat Dinah Pfizenmaier (Germany) 5-7 6-3 6-1
3-Paula Ormaechea (Argentina) beat Kiki Bertens (Netherlands) 7-5 7-5
5-Kurumi Nara (Japan) beat Hsieh Su-Wei (Taiwan) 6-3 6-2
Anna-Lena Friedsam (Germany) beat Nadiya Kichenok (Ukraine) 6-2 6-3