Neymar slotted home a penalty after Gabriel Jesus was brought down in 71 minutes.
The Palmerias forward doubled their lead 17 minutes later when he cleverly flicked home a cross from full back Marcelo.
He got his second and Brazil's third in injury time when he completed a counter attack with a curling shot from the edge of the box.
"You don't get a better debut than that do you?" said Gabriel Jesus, the 19-year old who will join Manchester City at the end of the year. "I am very happy to score on my debut not just for the goals but for the debut and the team."
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The win was Brazil's first in Quito for 33 years.
Although Ecuador started well, tormenting Brazil down the flanks, there were few chances for either side in a scrappy first half.
Brazil came into the game in the second period and were helped when Juan Carlos Paredes got a red card just moments after the visitors had opened the scoring.
The result lifts Brazil into fourth place in the 10-team South American group, though there are still two more games to come later on Thursday.
Group leaders Uruguay face Argentina in Mendoza, while Copa America champions Chile play Paraguay in Asuncion.
In earlier matches, Colombia overcame Venezuela 2-0 in Barranquilla, while Bolivia's new coach Angel Guillermo Hoyos got his reign off to the perfect start with a 2-0 win over fellow strugglers Peru.
James Rodriguez got Colombia's opener just a minute into the second half before Macnelly Torres sealed the win 10 minutes from time.
Wilker Angel and Rolf Feltscher were sent off for Venezuela but goalkeeper Daniel Hernandez saved his side from further embarrassment by stopping second-half penalties from both Carlos Bacca and Rodriguez.
In La Paz, Bolivia's Pablo Escobar put the home side ahead in 37 minutes with a spectacular free kick from long range and then Ronaldo Raldes doubled their lead with three minutes remaining when he headed home a corner kick.
The top four teams in the group qualify automatically for Russia 2018 and the fifth-placed side goes into an intercontinental play off.
(Reporting by Andrew Downie; Editing by Greg Stutchbury)
