With 20 minutes gone, Coutinho gathered a 25-yard clearance from goal and curled a shot into the far corner off the post to give the five-time world champions the lead.
However, Switzerland, who had looked toothless throughout the first half, got back on level terms five minutes into the second half.
Zuber rose unmarked to head home a corner kick.
Coutinho had Brazil’s best chance after the break but shot wide.
The result means that Serbia lead Group E after beating Costa Rica 1-0 earlier in the day.