The Central Americans must beat Switzerland in their final group game on Wednesday and rely on Ecuador to defeat France if they want to progress at the tournament.
Despite their chances of making the last 16 looking slim, Costly believes that Honduras are capable of getting through.
"Hope is the last thing you lose, as they say," he told FIFA.com. "But we know it'll be a very tough match against the Swiss because we know their game well.
"We're still hopeful and we've got faith that everything will turn out well, that's what we're clinging on to."
Costly has scored 32 goals in 71 appearances for Honduras since he made his debut back in 2007.