Both sides were poor in the first period but the game briefly came alive five minutes into the second half when Coutinho slotted home a VAR-assisted penalty.
Coutinho grabbed a second with a close-range header three minutes later before Everton rifled home a third from outside the box with five minutes remaining.
Wearing white shirts for the first time in more than 60 years, the home side huffed and puffed but rarely threatened and even after they went 2-0 up, they looked devoid of ideas and urgency.
Brazil, who started with Roberto Firmino up front and Fernandinho in midfield in place of the injured Arthur, failed to inspire the strangely subdued crowd of just 46,342 at Sao Paulo's Morumbi stadium.
"Pretty tough game. We were a little nervous because it is the first game," Coutinho said. "But when we scored the first one, we started to play properly. We the stayed focused and made the right decisions both technically and tactically."
The scoreless first half prompted boos from the faithful in the stands, which Coutinho said was "part of the game."
The fans want us to play well," the Barcelona star said. "We need to kind of shield ourselves and remain focused on the main goal. It's important to be locked in in order to play well."
Richarlison said it was understandable for the fans to be frustrated as Brazil seeks to win their first Copa America title since 2007.
"It's normal. The fans want to see goals but we kept our cool," Richarlison said. "The first half had a little bit of nervousness, but in the second we could made good passes and scored the goals easily. Now we have to focus on the next match to qualify for the next round."
Brazil's next match is against Venezuela in Salvador next Tuesday, which captain Dani Alves said will have a more festive atmosphere.
"[Playing in Salvador] will be different. They miss the [national team]," the Paris Saint-Germain defender said. "It will be more exciting than here."
The match was the first of this year's tournament, which is being held in Brazil for the first time since 1989. All 10 South American teams are participating along with guest nations Qatar and Japan.
Bolivia, who last won the title in 1963, rarely threatened at the other end.
Peru play Venezuela in Rio de Janeiro on Saturday in the second Group A game.
Information from Reuters was used in this report.