The 23-year-old scored with a shot from outside the box that was deflected by a defender to the right post in the 30th minute.
Five-time world champions Brazil lost consecutive matches to Uruguay, Colombia and Argentina at the end of last year and were in sixth place in the standings after six rounds. The Brazilians now have 10 points and are in fourth.
The Ecuadorians had a five-game unbeaten streak snapped and remained with eight points in sixth place, the last direct qualifying spot from South American for the World Cup which will be played in the United States, Mexico and Canada.
"We needed this win, doesn't matter if it was ugly or not," Rodrygo told Brazilian TV Globo.
"I'm happy with the victory and to have scored, and I hope it helps us getting better and progressing to the level that we want to reach moving forward."
Earlier on Friday, striker Luis Suarez retired from his national squad as Uruguay settled for a 0-0 draw against Paraguay at the Centenario stadium in Montevideo.
Luis Suarez, accompanied by his children, waves to fans after his final game for Uruguay. (AP PHOTO)
The team managed by Marcelo Bielsa that finished in third place at the last Copa America sit second with 14 points, four fewer than leaders Argentina, the current continental and world champions.
The 37-year-old Suarez had announced that the match against Paraguay was his last in Uruguay colours. The former Liverpool and Barcelona star retired with 69 goals scored from 143 appearances with the national team.
Uruguay played without five players from their starting line-up, including striker Darwin Nunez, who were suspended for their involvement in clashes with fans following the defeat to Colombia in the continental showpiece in July.
In the late game, Peru and Colombia drew 1-1 with Alexander Callens and Luiz Diaz grabbing the goals.
Colombia remain third (13) and Peru (3) bottom on the 10-team table after seven rounds.
With AP & Reuters.
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