The nine-time winners had 74% of possession and 19 shots but could not break through in their Group D opener.
Marquinhos had the ball in the net in the 30th minute after he tapped in at the back post following a Raphinha free-kick but the Video Assistant Referee ruled it out for offside.
Brazil boss Dorival Jr brought youngsters Endrick, who has signed for Real Madrid, and Savio off the bench but they could not find a winner.
Lucas Paqueta hit the woodwork just after the hour mark and spurned two late chances from good positions at the SoFi Stadium in Hollywood Park.
The West Ham United midfielder acknowledged his side must improve when they play Paraguay on Saturday (02:00 BST).
“We played a great game, we fought until the end, I had three chances myself, we were frustrated not to win, but we leave with confidence,” Paqueta said.
“We played our game, we have to improve our finishing, myself in particular, so that we can win.”
Costa Rica coach Gustavo Alfaro made a film analogy to emphasise how much his side had been written off before the match had started.
“We were Bruce Willis in ‘The Sixth Sense’, where the only one that did not know he was dead was him, right?” he said.
“Well, everyone thought we were dead before the movie started in our case.”
James rolls back the years
Colombia took the early initiative in Group D with a 2-1 win over Paraguay as attacking midfielder James Rodriguez produced an eye-catching performance.
The 32-year-old, who has played just a handful of minutes for Brazilian club Sao Paulo in the two months leading into the Copa America, had a hand in both of Colombia’s goals.
Rodriguez flighted a pinpoint cross to the far post that picked out Daniel Munoz to head in Colombia’s opener in the 32nd minute.
The former Real Madrid and Everton playmaker provided another assist just before half-time with a free-kick that was glanced in by Crystal Palace midfielder Jefferson Lerma.
Brighton’s Julio Enciso pulled a goal back for Paraguay in the 69th minute with an angled volley but Colombia held on to claim the points.
Colombia play Costa Rica in their second group game on Friday (23:00 BST) before facing Brazil on 2 July.