Sweden won 1-0 against South Africa in the opener, before host nation Brazil beat China 3-0.
Canada's Janine Beckie scored after just 20 seconds against Australia, the fastest goal in Olympic women's football history, as they won 2-0 at the Corinthians Arena in Sao Paolo.
The USA, who are attempting to win a fourth consecutive Olympic gold medal, started their campaign with a 2-0 win over debutants New Zealand.
Fifa women's player of the year Carli Lloyd and Alex Morgan scored in each half at Mineirao stadium in Minas Gerais, as the US retained their unbeaten record in 2016.
In the final match of the day, France cruised past Colombia 4-0. An own goal from Carolina Arias opened the scoring, followed by a header from Eugenie Le Sommer, and free-kicks from Camille Abily and Amel Majri.
More than 11,000 athletes will compete in the first edition of the Olympic Games to be held in South America.
There are 42 Olympic sport disciplines, with 306 events over the course of 19 days of competition.