The result left Serbia atop the group, as they beat Costa Rica 1-0 earlier in the day.
It was the same story as defending champions Germany, who were stunned 1-0 by Mexico in their opening game and now have work to do to make it through the group stage.
Brazil went all out in pursuit of a victor and came agonisingly close to finding it but had to settle for a point and joined fellow tournament favourites Germany, Spain and Argentina in failing to win their first game.
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The bright spot of Brazil's performance was Coutinho's trademark strike. Despite making 20 attempts toward Sommer's goal (including four shots on target and another nine off frame), Brazil created few genuine scoring chances. In order to do that, they'll first need to get past Switzerland in their Group E opening match.
Xherdan Shaqiri whipped a sensational corner into the six-yard box which found Zuber in space despite five Brazilian defenders being within yards of him.
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And deep into stoppage time Schar's outstretched leg turned a goalbound effort from Silva wide as Switzerland held out for a vital point.