Beatriz scored the lone goal for the Brazilians, who had yielded just one goal in their previous five matches in the Olympic tournament.
"I was not going to leave this tournament without a medal around my neck," said a tearful Sinclair after the match.
Canada defeated France 1-0 four years ago to win bronze. The Brazilian women won silver in 2004 and 2008, finishing sixth at the 2012 Games. They placed fourth in 1996 and 2000.
Germany beat Sweden 2-1 later in the day to win their first-ever gold medal in women's soccer.