The Central American nation made the quarter-finals of this year's World Cup and looked in good form as they went 4-1 up with 50 minutes on the clock.
Oman pegged them back with a pair of quick goals, but a final half-hour saw much of the quality seep out of the game and the hosts were unable to draw level.
Paulo Wanchope's Costa Rica side went ahead as early as the fifth minute, Bryan Ruiz whipped a corner in from the right and Alvaro Saborio headed home as Oman goalkeeper Ali Al Habsi flapped at the delivery.
Emad Al Hosaini was at the heart of the hosts' response, though, crashing a 25-yard effort against the post before feeding Raed Ibrahim Saleh to jink past Yeltsin Tejeda and slot beyond Esteban Alvarado.
Costa Rica went in at the break ahead when John Jairo Ruiz hit an effort that Al Habsi was unable to keep out after good work down the right from Bryan Ruiz.
Oman could not have started the second period much worse as a string of defensive lapses allowed the visitors to seize control of the game.
First, Juan Bustos slammed home via the post after Junior Diaz had bustled his way into the penalty area past an Oman defence that passed up at least four chances to clear.
And Bustos turned provider of sorts for number four as his overhit throughball should have been easily dealt with by defender Abdul Sallam Amur or Al Habsi, but they each left the ball for each other and substitute David Ramirez prodded home.
However, the hosts roared back and two goals within five minutes had them back in the contest.
Mohammed Al Seyabi's thunderous 25-yard strike arrowed in past Alvarado and Costa Rica were reeling as a swift counter released Qasim Said on the left, who showed composure to pick out Al Hosaini to drill home.
Paul Le Guen's men found joy on the flanks, but a series of enticing deliveries were just off the mark or strikers were out of position and Costa Rica ultimately held on for the win.
