Radwanska made short work of Sweden's Johanna Larsson in a 6-0 6-1 victory that took 44 minutes.
[Australian Open - Watch LIVE on British Eurosport]
"It's really hot today," Radwanska said. "So I'm really happy that I play short and good and I closed that match pretty quickly."
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[Total Australian Open coverage on Eurosport Player]
Wimbledon champion Petra Kvitova also enjoyed a straightforward win as she beat Germany's Mona Barthel 6-2 6-4 - despite Barthel pushing her all the way during one marathon game, which lasted 16 minutes.
"It's my best (performance) in the past three years, so I have to be happy with that and I'm looking forward to the next match," said Kvitova, adding that she always believed she could turn that epic game around.
"I was losing 40-15 (but) I knew I have a good serve and could help myself."But 16 minutes, it's almost a record for me. In the end I did it, but Mona played very well in the second set.”
France's Alize Cornet faced a tougher battle, needing over two and a half hours to beat Denisa Allertova (Czech Republic) 6-4 6-7(2) 6-2.
The second set alone lasted 72 minutes as the 19th seed made it through.
Venus Williams continued her encouraging form at the Australian Open as she beat fellow American Lauren Davis 6-2 6-3.
The star's struggles with Sjogren's syndrome, an auto-immune disease, have made every match she plays a lottery, but the 18th seed had little trouble accounting for Davis in straight sets.
Thirty-four year-old Venus was dumped from the first round last year but now finds herself on a seven-match winning streak after clinching the Auckland Open title in the lead-up.
Venus faces a tough third-round match against 33rd-ranked Camila Giorgi, a feisty Italian counter-puncher, who has broken into the top 50 with a bullet.
The American said her battles with health problems had not lessened her desire for more titles after more than 20 years as a professional.
"Honestly, I think I understand the game a lot more," she said. "Even when I'm not playing as well, I think I'm able tactically to be more strategic than even, let's say, Venus of 2000.
"So that's one of the beauties of continuing to play as you get a lot of the years under your belt."
Second round results
Victoria Azarenka (Belarus) beat 8-Caroline Wozniacki (Denmark) 6-4 6-2
25-Barbora Zahlavova Strycova (Czech Republic) beat Chang Kai-Chen (Taiwan) 6-1 7-5
Madison Brengle (U.S.) beat Irina Falconi (U.S.) 6-1 6-3
4-Petra Kvitova (Czech Republic) beat Mona Barthel (Germany) 6-2 6-4
19-Alize Cornet (France) beat Denisa Allertova (Czech Republic) 6-4 6-7(2) 6-2
Madison Keys (U.S.) beat 29-Casey Dellacqua (Australia) 2-6 6-1 6-1
26-Elina Svitolina (Ukraine) beat Nicole Gibbs (U.S.) 7-6(3) 7-6(6)
11-Dominika Cibulkova (Slovakia) beat Tsvetana Pironkova (Bulgaria) 6-2 6-0
30-Varvara Lepchenko (U.S.) beat Ajla Tomljanovic (Croatia) 6-1 7-6(1)
Camila Giorgi (Italy) beat Tereza Smitkova (Czech Republic) 6-1 6-4
1-Serena Williams (U.S.) beat Vera Zvonareva (Russia) 7-5 6-0
Timea Bacsinszky (Switzerland) beat Anna Tatishvili (U.S.) 6-7(5) 6-3 6-2
24-Garbine Muguruza Blanco (Spain) beat Daniela Hantuchova (Slovakia) 6-1 1-6 6-0
18-Venus Williams (U.S.) beat Lauren Davis (U.S.) 6-2 6-3
6-Agnieszka Radwanska (Poland) beat Johanna Larsson (Sweden) 6-0 6-1
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