Top-seed Stosur took almost 1- 1/2 hours to overcome qualifier Hradecka, coming back from 4-1 down in the first set to set up a second-round meeting with compatriot Jarmila Groth.
Eighth-ranked Andy Roddick, the men's second seed, served five aces and one double fault, spending 72 minutes on court in his 6-3, 6-2 win over Australian wild card Marinko Matosevic.
Roddick dominated on serve against his 138th-ranked opponent, winning 77 percent of service points and taking advantage of four of six break points.
He said later he felt under no pressure to defend the Brisbane title he won last year when he beat Radek Stepanek in the final.
"More so the pressure is always to use this tournament to get me ready for Melbourne. It worked last year so I'm going to do it again this year," Roddick said.
Stosur served eight aces and converted five of nine break points, steadily improving as the match progressed.
She looked both nervous and rusty at the start of her match with 110-ranked Hradecka, her first since bowing out in the semifinals of the WTA Championships in Doha in October. The 26-year old Stosur struggled to get on top of Hradecka, falling behind in the first set before recovering her touch and confidence to dominate from the baseline.
"Today is a good sign that I can handle things a bit better," Stosur said. "I was really pleased with the way I was able to fight my way back and get through that first set.
"One of the reasons I was able to do that today was I didn't panic, didn't start freaking out thinking I had to play unbelievably to get back. I just took it point by point.
"Last year I was quite disappointed with the way I started the summer. Overall I think my game has improved not just for Australian hardcourts but in general. I am a better player than this time last year."
Stosur said she was ready to carry the weight of expectation that will fall on her at this month's Australian Open where she is seen as Australia's best title chance in 30 years.
"All the top players go through it, it's just a matter of dealing with it, playing things out in your mind so nothing gets you by surprise," she said.
"The more you experience it the more you will feel better about it."
Second-seed Shahar Peer of Israel beat Australian wild card Sophie Ferguson 6-4, 6-2 to also progress to the second round on Monday. But third-seeded Russian Russian Nadia Petrova bowed to Petra Kvitova of the Czech Republic 7-6, 6-3.
French fourth seed Marion Bartoli brushed aside American qualifier Vania King 6-1, 6-2.
World No. 30 Daniela Hantuchova withdrew shortly before her first round match on Monday and is now racing the clock to recover from an Achilles tendon tear in time for the Australian Open.
In the men's draw, top-seeded Robin Soderling will open his campaign Tuesday against qualifier Ryan Harrison. The world No. 5 has yet to win a hardcourt title.
"I don't know why, because it is a surface that should be very good for me," Soderling said. "I have never had the good results but hopefully that will change for me this year."
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