The British number one is no fan of playing on clay and had suffered an embarrassing straight sets defeat to world number 219 Fanny Stollar in her previous match in Charleston last month.
But Konta was in much more impressive form as she battled through two tight sets to see off her fellow Wimbledon semi-finalist 6-3 7-5 in two hours.
Konta made a flying start to break Rybarikova, who is also by no means a clay specialist, and move 3-0 up in the opening set.
She maintained her momentum through a series of lengthy service games to establish a one-set lead, but was broken early in the second set before fighting back to level.
A second crucial break enabled Konta to wrap up victory and book a second round match against American Bernarda Pera, who beat qualifier Aryna Sabalenka in straight sets.
Sharapova ends losing run
Maria Sharapova ended her four-match losing streak with victory in the first round of the Madrid Open on Sunday, and insisted she is determined to claw her way back to the top of her sport.
Sharapova has struggled since her return from a drugs ban and fell outside the world's top 60 in recent months but she proved too good for Romania's Mihaela Buzarnescu, winning 6-4 6-1 in 82 minutes.
Sharapova told a post-match press conference: "Do I want to be ranked 60, 70 in the world? No, I don't.
"Do I want to be losing in the first round? Absolutely not. That's why I'm still here, because I'm not satisfied with those things.
"I keep looking and getting better and working on things, making adjustments and not being stubborn about things I believe will make me better."
Another former world number one, Victoria Azarenka, earned her first victory on clay in more than two years as she swept aside Aleksandra Krunic of Serbia 6-3 6-3 in another first round clash.
Top seed Simona Halep needed just 51 minutes to beat Ekaterina Makarova of Russia 6-1 6-0, nor was second seed Caroline Wozniacki unduly detained as she saw off Daria Gavrilova 6-3 6-1.
Third seed Garbine Muguruza beat Shuai Peng of China 6-4 6-1, and another former Wimbledon champion Petra Kvitova continued her recent good form with a 6-1 6-2 success against Laura Tsurenko.
Other winners on a busy Sunday of first round action included Sloane Stephens, Sam Stosur, Donna Vekic, Ashleigh Barty and Monica Puig.
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