Davies excelled throughout an emphatic 4-0 win over Pep Guardiola's side at Goodison Park and the 18-year-old crowned his display with a brilliant solo goal for Everton's game-sealing third.
Koeman's side continue their Premier League campaign at Crystal Palace on Saturday and, although delighted with their progress, the Dutchman had a calm yet stern message for Davies and fellow youngster Ademola Lookman – Everton's new signing from Charlton Athletic who completed City's misery with a debut goal.
Asked how Davies was handling being thrust into the spotlight, Koeman said: "I hope good because it's only one game. It's only three weeks where he is really involved [in the first team].
"He has to deal with that attention. I think it's positive. It's a young player who took his chance in the first team
"The manager and the rest of the coaching staff, we have to deal with that situation with young players – even with Lookman, all the attention he got after his first goal.
"We must keep their feet on the ground. That is the most important.
"If we see young players are not focused on the game because they think they are now the big men in Everton then we will do something that keeps their feet on the ground."
Koeman remains keen to bring in further January additions, alongside Lookman and Morgan Schneiderlin, while stating he would be happy to "find a solution" for Gerard Deulofeu and Phil Jagielka as they languish outside of the first team.
"I hope for more players coming in," he added.
"We are working hard to make the team stronger. But we are looking for quality, not just numbers.
"The relationship with [director of football] Steve Walsh is good. We talk daily about players and about scouting."
Everton left-back Leighton Baines is winning his battle to overcome a rib injury in time for the trip to Selhurst Park, while goalkeeper Maarten Stekelenburg is available once more.