French Open champion Jelena Ostapenko was upset 6-4 6-2 in the first round by unseeded Serbian Aleksandra Krunic and Agnieszka Radwanska, seeded 10th, was another to suffer an early exit, losing 6-4 6-4 to German Julia Gorges.
Spanish fourth seed Muguruza, fresh off a quarter-finals loss to eventual winner Elina Svitolina at Toronto last week, needed only 65 minutes to earn the straight sets win over Haddad Maia and book her spot in the round of 16.
After trading breaks with Haddad Maia over the course of the first four games, Muguruza found another gear and went on to win the final 10 games of the match.
"I felt like a lot of games were super tight. She's super tall, big serve with big shots. It was a strange result, because the match was pretty good," said world number six Muguruza.
"I feel like yesterday I was playing on the grass, and now I'm here. I'm happy to be in the third round here."
Muguruza now awaits the winner of Wednesday's clash between American 16th seed Madison Keys and Russian Daria Kasatkina.
Romanian Halep, the 2015 Cincinnati runner-up, fell behind early but reeled off four consecutive games, including two love service holds, to clinch the opening set and remained in control the rest of the way.
Halep will next face Latvia's Anastasija Sevastova, who earlier rallied to beat China's Peng Shuai 3-6 6-4 6-2.
Williams was aggressive from the start and only got better as the match went on, stepping in and finishing rallies at the net.
Fashion-conscious Williams' biggest struggle on the court may have been with her jewellery after her hoop earring fell off at one point and later got tangled with her necklace.
The win set up a second round match-up against Australian Ashleigh Barty, who beat American Varvara Lepchenko earlier on Tuesday.
(Reporting by Rory Carroll in Los Angeles; Additional reporting by Frank Pingue in Toronto; Editing by Peter Rutherford)
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