The clock had ticked past 90 minutes, and the five-times champions looked to be edging to a second disappointing draw, when Marcelo floated the ball in from the left and substitute Roberto Firmino headed across to Gabriel Jesus.
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 metres out for his second goal of the tournament.
The result leaves Brazil with four points from two games after they drew 1-1 with Switzerland in their opening match, and they provisionally rise above Serbia to top Group E.
Serbia play the Swiss later on Friday, while Costa Rica are out of the tournament with defeats in both games so far.
Neymar added the second with practically the last kick of the match, volleying home following a quick counter-attack as Costa Rica pushed for an equaliser.
With the game still scoreless 13 minutes from time, Neymar collapsed under a soft challenge in the box and thought he had won a penalty for his misfiring side.
Dutch referee Bjorn Kuipers pointed to the spot before heeding the protests of the Costa Rican players and consulting the video assistant referee (VAR). Replays showed there was minimal contact and Kuipers reversed his decision as Brazil's frustration reached fever pitch.
Time and again they hurled themselves at the Costa Rican goal only to be repelled until their opponent's resistance finally broke.
Neymar, overcome by emotion on the whistle, covered his face with his hands as tears streamed down his cheeks.
That was an apt reflection of how tense the match had been, with Costa Rica's flat-back five keeping Neymar on the periphery, as Johan Venegas and Cristian Gamboa man-marked him out of the game.
"We had a plan, strategy, idea to control the opponent with and without ball," Costa Rica coach Oscar Ramirez said.
DOMINATING POSSESSION
Brazil had 71.7 percent of the possession, but for all their dominance found it hard to carve out clear chances.
Marcelo's tame shot straight at Navas just before halftime was the first effort on target from either side, while Celso Borges spurned the best chance of the half for Costa Rica when he ran on to Gamboa's cut back but shot wide.
Brazil went into the break with the boos and whistles of their fans ringing in their ears but emerged with a renewed sense of purpose, with Jesus heading against the bar four minutes after the restart and Coutinho sending the follow-up effort narrowly wide.
The finish was frantic as Brazil attacked with more desperation, until Coutinho rode to their rescue.
"We deserved the win," the Barcelona player said. "The ball came free and Gabriel shielded it well and I managed to score. The most important thing was winning."
(Reporting by Simon Jennings in St Petersburg; Editing by Christian Radnedge)
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