The result left the CONCACAF champion in third place in Group A with three points, trailing Ukraine by one point and just behind host Thailand on goal difference. Ukraine defeated Thailand 5-3 in another match Sunday.
Paraguay jumped ahead after 1:41 when Costa Rican keeper Jairo Toruna had the final touch to knocked Fabio Alcaraz's kick-in into his own net. Alcaraz then doubled the lead nine minutes later, sprinting past Diego Zuniga and scoring with a drive from the right touchline.
Zuniga began Costa Rica's rally two minutes into the second half, volleying Cubillo's diagonal pass into the penalty area past Paraguay keeper Carlos Espinola.
Costa Rica equalized in the 25th minute when Cubillo intercepted a ball at midcourt, worked his way around the left and sent a cross in front of goal that struck Luis Navarrete's hip, deflected off Paraguayan defender Gabriel Ayala onto the crossbar and dropped for Navarrete to prod in at the left post.
Cubillo scored his first less than two minutes later to give Costa the lead, Navarrete added a second a minute after that and Cubillo tacked on a second in the 31st with Ayala wearing the goalkeeper's jersey to put the Ticos ahead 5-2.
Walter Villalba pulled Paraguay within two in the 37th, but Cubillo restored the margin with 16 second left with another empty-net goal.