Cameroonian goalkeeper Onana was the hero of the hour for Man United in their 1-0 Champions League win over Copenhagen two weeks ago, saving a late penalty from Jordan Larsson.
Onana - who has been criticised for several errors in the early days of his Man United career - was immediately mobbed by his teammates, who also paid tribute to his contributions on social media.
However, Garnacho found himself in hot water after sharing a photo of Red Devils players celebrating with Onana accompanied by two gorilla emojis, leading to the FA to investigate the matter.
Onana himself said that the incident had been blown out of proportion, claiming that Garnacho meant to symbolise "power and strength" with his use of the icons, while Erik ten Hag was also not concerned about the prospect of the Argentine being reprimanded.
Following the FA's investigation, the governing body were satisfied that Garnacho did not use the emojis in banned faith and will not be taking any action against the teenager.
"We sought the player's observations as part of our investigation, and he explained that the use of two gorilla emojis was intended to highlight the strength and power of his team-mates - specifically Andre Onana and Harry Maguire - following the pivotal roles that they played in Manchester United's win over FC Copenhagen," the FA said in a statement.
"We are satisfied with Alejandro Garnacho's explanation and the context that it provides, so we will not be issuing disciplinary proceedings on this occasion.
"We also intend to liaise with Kick It Out and the PFA to discuss other potential issues around the use of emojis and similar forms of messaging, and education that can be provided about them."
Garnacho has avoided punishment four years after Manchester City's Bernardo Silva was fined £50,000 and hit with a one-game ban for a tweet about former teammate Benjamin Mendy, comparing the left-back to a character on an Iberian brand of chocolate.
Both Mendy and Silva insisted that the post was nothing more than a joke between two friends, but he was deemed to have posted content "insulting by reference to race, colour and ethnic origin in a way that unquestionably brings the game of football into disrepute".
Garnacho came off the bench for the final 27 minutes of his side's victory over Copenhagen two weeks ago, and the 19-year-old will expect to be involved in some capacity during Wednesday's return fixture.
The Argentina international has one goal to show from 14 matches in a Man United jersey this season, but he has only started three Premier League matches in 2023-24 so far.
Meanwhile, Onana's clean sheet in Saturday's 1-0 Premier League win over Fulham was his fifth in his first 16 matches for Man United, during which time he has also let in 26 goals.