The US won the World Cup qualifier thanks to Clint Dempsey's goal but the match was played in awful conditions in Denver, with snow engulfing the pitch.
Costa Rica were unhappy with the decision to play the match, which only went ahead after plows and shovels were used to clear the penalty areas and centre cirle.
Costa Rican Football Federation treasurer Rolando Villalobos said: "What happened... has no precedents, since it not only went against the sporting spectacle, but also against the physical integrity of the officials and players."
FIFA said on Monday that Costa Rica had requested that the game be replayed.
"We can confirm that FIFA has received a letter from the Costa Rica FA regarding last Friday's World Cup qualifier," said soccer's governing body in a statement. "FIFA will now analyse the content of the letter and next steps will be determined in due course."
Furious Costa Rica coach Jorge Luis Pinto said the match was an "embarrassment to football" after his side lost 1-0 in Denver on a snowy field with a covering that became deeper as the game wore on.
The Costa Rica Football Federation (Fedefutbol) said in a statement on its website that there were four parts to its protest.
It said the conditions were a threat to "the physical integrity" of the players and said stadium officials had come on to the pitch to clear the snow while the ball was in play.
Furthermore, the pitch markings "disappeared" and the "the movement of the ball became impossible due to the quantity of snow on the pitch."
Fedefutbol also demanded sanctions for "all the officials who were involved in the decision to keep playing this game."
"The complete opposite happened in Europe where the snow forced the match between Northern Ireland and Russia in Belfast to be postponed," it added in a separate statement.
"We don't want to create false expectations but we are fighting for rights which we think we violated during this match," said Fedefutbol treasurer Rodolfo Villalobos.
Costa Rica could retaliate by staging the return fixture in the Saprissa stadium, a notoriously hostile arena where the fans are close to the pitch, rather than the new National Stadium, which has a running track around the field.
The defeat left Costa Rica bottom of the six-team final stage in the CONCACAF region qualifiers with one point from two games. The Ticos host Jamaica in San Jose on Tuesday.