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Osaka has struggled since splitting with her coach Sascha Bajin in February while an abdominal injury forced her to withdraw in Stuttgart eight days ago, creating doubts about her challenge at the French Open later this month.
But there was little sign of physical problems during a testing 6-2, 7-6 (8/6) victory over Cibulkova and afterwards, Osaka said: "I feel good. I don't have much pain in my abs so that's always a good sign."
The Japanese world number one will now face Spain's Sara Sorribes Tormo in round two.
Osaka looked set for a more comfortable passage after a dominant opening set but she was stretched by the battling Cibulkova in the second.
By the end, Osaka had hit 43 winners and eight aces while world 33 Cibulkova conjured up 23 winners.
"I was also kind of thinking, like, there is nowhere else I'd rather be and honestly I train for these moments so I have to give it 100 percent," said Osaka after her first main draw win in Madrid.
"I feel like there's an adjustment period for me on clay and it took a few years.
"But now I'm more comfortable. I'm not sure if it's because I had a really long training block before I came to start the season, but yeah, it feels good."
Osaka's pair of Grand Slam successes have come on hard, at the US and Australian Opens, but she is yet to find her best on clay.
If the abdominal injury has healed, Osaka can feel encouraged by her performance against Cibulkova, one of the tour's gutsiest competitors even if the Slovakian is also enduring a difficult start to 2019.
Meanwhile, French Open champion and two-time Madrid winner Simona Halep brushed past Russian qualifier Margarita Gasparyan 6-0, 6-4.
"It was a good match, even if the first set was fast. The second one was tougher, and I feel happy that I could win this match," said third-seeded Halep.
Former world number one Caroline Wozniacki's Madrid Open lasted just minutes when she retired at 0-3 down to France's Alize Cornet suffering with a back injury.
Wozniacki, the 2018 Australian Open champion, has been battling rheumatoid arthritis in recent months.
Fourth seed Karolina Pliskova saved five match points to see off highly-rated Ukrainian teenager Dayana Yastremska 5-7, 7-6 (7/5), 6-3.
Unseeded Pauline Parmentier of France defeated Ukrainian sixth seed Elina Svitolina, who had been struggling with a knee injury, 6-4, 7-6 (8/6).
US 13th seed Madison Keys was also a surprise first round loser, going down 3-6, 6-4, 6-1 to Sorana Cirstea of Romania.
However, there were no such problems for former US Open champion - and recently engaged - Sloane Stephens as the eighth seed defeated Polona Hercog of Slovenia 6-2, 7-6 (7/4).
Stephens, who announced her engagement to US football international Jozy Altidore last week, goes on to face former world number one Victoria Azarenka.
In one of the few ATP matches played on Sunday, Australia's Nick Kyrgios was a 7-6 (7/4), 6-4 loser to Germany's Jan-Lennard Struff, a quarter-finalist in Barcelona last month.
Results from the ATP and WTA tournaments in Madrid on Sunday (x denotes seeding):
Men
First round
Jan-Lennard Struff (GER) bt Nick Kyrgios (AUS) 7-6 (7/4), 6-4
Felix Auger-Aliassime (CAN) bt Denis Shapovalov (CAN) 6-2, 7-6 (9/7)
Women
First round
Naomi Osaka (JPN x1) bt Dominika Cibulkova (SVK) 6-2, 7-6 (8/6)
Sara Sorribes (ESP) bt Lara Arruabarrena (ESP) 6-4, 3-6, 6-1
Anna Karolína Schmiedlová (SVK) bat Elise Mertens (BEL) 7-5, 7-6 (7/5)
Belinda Bencic (SUI) bt Alison Van Uytvanck (BEL) 6-4, 6-3
Kateryna Kozlova (UKR) bt Zhang Shuai (CHN) 6-3, 6-2
Karolína Pliskova (CZE x5) bt Dayana Yastremska (UKR) 5-7, 7-6 (7/5), 6-3
Simona Halep (ROM x3) bat Margarita Gasparyan (RUS) 6-0, 6-4
Carla Suárez (ESP) bt Maria Sakkari (GRE) 6-3, 6-3
Viktoria Kuzmova (SVK) bt Julia Goerges (GER x16) 7-5, 6-4
Danielle Collins (USA) bt Vera Zvonareva (RUS) 6-1, 6-4
Pauline Parmentier (FRA) bt Elina Svitolina (UKR x6) 6-4, 7-6 (8/6)
Sloane Stephens (USA x8) bt Polona Hercog (SLO) 6-2, 7-6 (7/4)
Zheng Saisai (CHN) bt Wang Yafan (CHN) 7-5, 7-6 (7/3)
Alizé Cornet (FRA) bt Caroline Wozniacki (DEN x11) 3-0 - retired
Donna Vekic (CRO) bt Wang Qiang (CHN x15) 7-5, 6-4
Kristýna Plíšková (CZE) bt Marta Kostyuk (UKR) 3-6, 6-3, 6-2
Petra Martic (CRO) bt Garbiñe Muguruza (ESP) 7-5, 7-6 (7/2)
Angelique Kerber (GER x4) bt Lesia Tsurenko (UKR) 6-3, 6-2
Mihaela Buzarnescu (ROM) bat Ajla Tomljanovic (AUS) 6-3, 6-2
Sorana Cirstea (ROM) bt Madison Keys (USA x13) 3-6, 6-4, 6-1
Kristina Mladenovic (FRA) bat Barbora Strýcová (CZE) 6-3, 6-1
