The Caribbean side had the first effort on goal through captain Rudolph Austin, but he failed to test the keeper from long range.
© Getty Images
However, his side took the lead four minutes later when a long throw travelled to the back post where Garath McCleary slammed the ball into the roof of the net.
However, Costa Rica equaliser just after the half-hour mark when Roy Miller nodded in from a few yards out following a set-piece move.
The World Cup quarter-finalists then took the lead a few minutes later as David Ramirez latched on to the end of a long pass over the top of Jamaica's defence and he lofted it over the keeper.
Jamaica restored parity shortly after the break though, as Jobi McAnuff side-stepped one challenge before prodding it low into the back of the net.
However, neither side could find a winner and start Group B with a point.