The game was held at the 43,500-seater Rostov Arena on Sunday.
Brazil's Philippe Coutinho broke the deadlock in the 20th minute after curling a beautiful right-footed shot from the edge of the penalty area, which went in off the post.
Switzerland's Steven Zuber leveled the score five minutes after half time with a fine close-range header.
Serbia defeated Costa Rica 1-0 in the other Group E match earlier in the day.
Brazil will next play against Costa Rica on Friday in St. Petersburg, while Switzerland will face Serbia on the same day in Kaliningrad.