Japan's four-time grand slam champion stumbled on match point as Bronzetti completed a 6-4 2-6 6-4 win before checking Osaka was OK as they approached the net for a post-match handshake.
The 55th-ranked Osaka was unseeded in Madrid for her first match in almost a month since reaching the fourth round of the Miami Open. Italy's Bronzetti will play fifth-seeded American Madison Keys in the second round.
The 26-year-old's win was her third against a former world No.1 this year.
Bronzetti beat Simona Halep in the Romanian's final career match in Cluj-Napoca and Victoria Azarenka at the Australian Open in January.
Kimberley Birrell was eliminated after a nearly two-and-a-half hour battle with Peyton Stearns, going down to the American 5-7 6-3 6-4.
New Aussie No.1 Daria Kasatkina, who has a bye to the second round as No.14 seed, and Ajla Tomljanovic are also in the 128-player draw for the WTA 1000 event.
Alexandra Eala will get a rematch with Iga Swiatek in the second round in Madrid after stunning the Polish player as a wildcard in Miami last month. Eala won her first-round match 6-3 6-2 against Viktoriya Tomova.
The 19-year-old Eala, a product of Rafael Nadal's academy, has climbed to No.72 in the WTA rankings after reaching the Miami Open semi-finals as a wildcard entry last month. She's the first Filipina to be ranked inside the top 100.
"Now that the dust has settled, I try not to think about Miami so much," she said.
"Of course, it comes to mind when everyone's asking, but in my mind I have my whole life to look back at those moments and I'll always have those memories. But the tour goes on, competition goes on, so right now I'm fully focused on Madrid."
There were also victories for Alex de Minaur's British fiancee Katie Boulter, New Zealand's Lulu Sun, Americans Caroline Dolehide and Hailey Baptiste, Spain's Jessica Bouzas Maneiro, Camila Osorio of Colombia and Czech Marie Bouzkova.
with PA
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