Costa Rica played an nearly ideal defensive game, holding off the Brazilians until the 90th minute, when Coutinho scored to shatter their hopes before Neymar made it 2-0 deep into extra time on Friday, reports Xinhua news agency.
In the second half, Brazil were more aggressive but Costa Rica still found a way to stop them in the final third by switching their 3-4-2-1 formation to a 5-4-1 whenever the Brazilians tried to pounce into their penalty box.
Jesus's first touch took the ball away from the defender and into the path of Coutinho who raced in to poke past goalkeeper Keylor Navas from six meters out for his second goal of the tournament.
Neymar thought he had a penalty with 13 minutes remaining when the referee awarded a spot-kick, then changed his decision after a video assistant referee (VAR) review and adjudging he dived.
The win puts Switzerland into second place in Group E with four points, the same as Brazil.
It came at the ideal time with Brazil set to face a tough Serbia squad and the confidence they gained with this win could help them get three more points against the European squad.
After an opening game defeat to Serbia, Costa Rica know that another loss to five-time world champions Brazil today could spell the end of their time in Russian Federation.
"It's not normal to cry in the second game of a World Cup".
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His sister, Genesis Collado-Feliz, claims that her brother ran into the bodega because he thought he would be safe in the store. After a few seconds, Guzman-Feliz was seen standing on the street corner with blood pouring from a wound in his neck.
Neymar's obsession with social media has undoubtedly opened him up to more criticism - and pressure - than ever but former players acknowledged even the most balanced individual is under massive stress in Russian Federation.
Brazil, too, need a win, if only to show that their surprise draw with Switzerland in their first game of the tournament was a one-off. "The second half, it was handsome, perfect".
Neymar let his emotions show after scoring for Brazil against Costa Rica, leading the FC crew to debate whether the display was over the top.
Brazilians watched their team's excruciatingly hard-fought win Friday over minnow Costa Rica in a cold sweat but it was Neymar's hot tears that really made the footballing-mad nation sit up.
That sets up a showdown between Nigeria and Argentina in Saint Petersburg on Tuesday.
"It's possible I'm going to score another two goals", said Musa, 25, who scored Friday in the 49th minute on Victor Moses' cross and again in the 75th minute.
In Russia, however, Costa Rica are already on the back foot having lost to Serbia in their opening game.
"It's the country with the most World Cups, so it's very different, but it's still the game, it's still enjoyable and it's still fun".