The highly organised Swiss hit back early in the second period from one of their few chances as Brazil's Casemiro and Miranda switched off and the unmarked Steven Zuber barely had to leave his feet to nod in Xherdan Shaqiri's whipped corner.
Brazil went all out in pursuit of a winner and came agonisingly close to finding it as Coutinho and Miranda lashed wide from inside the area while Neymar and substitute Roberto Firmino were denied by Swiss goalkeeper Yann Sommer.
It was also the first time Brazil had not won a World Cup opener since drawing 1-1 with Sweden in the 1978 tournament.
"I was expecting to get a victory so, of course, I'm not happy with the result," Brazil coach Tite told a news conference.
"At this stage of a World Cup you have to take advantage of a goal and see that your opponents are taking risks and you need to have faster transitions."
Tite did not complain that his side were denied a penalty when Manuel Akanji wrapped his arms around Gabriel Jesus and had little sympathy with claims some of his players had been fouled when Switzerland levelled.
"I can't coach a team who complains. You can interpret the penalty in any way but there's no way you can complain," he added.
Neymar's flamboyant blond 'spaghetti' haircut certainly caught the eye although he only gave glimpses of the spectacular dribbles and plays he is renowned for and which led Paris St Germain to shell out 222 million euros for.
Instead, Brazil's second most expensive player, Coutinho, pulled the strings, constantly finding right winger Willian and drifting striker Jesus with sweeping passes although Switzerland managed to deal with the threat.
Coutinho's whirlwind finish delighted the Rostov Arena, where yellow-shirted Brazil fans vastly outnumbered the red of the Swiss, although Tite's side failed to build further momentum, as Neymar was frequently stopped in his tracks by tactical fouls.
Brazil always looked in control until they ceded a corner and Zuber took full advantage of their collective lapse of concentration to level.
Switzerland coach Vladimir Petkovic was delighted with how his side followed his gameplan.
"I am very proud of my team's discipline, it was everything we talked about," he said.
"I hope that people will take notice of us now and take us seriously, sometimes there's a lack of recognition towards us and that's a pity because we played very well.Published: 2018-06-18 07:31:22