Muguruza, the 2016 French Open champion and 2017 Wimbledon winner, advanced to the second round at Flushing Meadows by beating Japan's Nao Hibino in straight sets.
The Spanish 10th seed needed 1hr 30min to complete a 6-4, 6-4 victory over Hibino, the world number 78.
After her win she immediately offered support to Navarro who said Tuesday she had been diagnosed with Hodgkin lymphoma and will require six months of chemotherapy.
Navarro had pulled out of the US Open last week for medical reasons.
"When we spoke a few days ago when she gave me the news, I was, I think, shocked, because I was expecting to see her in this tournament," said Muguruza.
"You know, she's such a nice woman, so sweet, so kind, so humble. When these things happen to these good people, I feel, like, so sad about it.
"I for sure will dedicate this win to her because I want her to feel that we are behind her, that I am behind her, and I will go and see her at some point when it is fine," added the Spaniard.
Muguruza will face Bulgaria's Tsvetana Pironkova in the next round. Pironkova advanced with a 6-2, 6-3 win over Russia's Liudmila Samsonova.
In other early women's matches, 16th seed Elise Mertens of Belgium cruised into the second round with a 6-2, 6-2 win over Germany's Laura Siegemund.
US hope Amanda Anisimova, seeded 22, also made sure of her place in the second round with a hard-fought win over Bulgaria's Viktoriya Tomova.
Anisimova prevailed 7-5, 7-5 in 1hr 27 minutes and now faces either Natalia Vikhlyantseva of Russia or compatriot Katrina Scott.