With Everton trailing 4-0 at half time, Davies was introduced by Allardyce and proved to be one of the few positives from the setback at the Emirates Stadium.
Davies has been given just 12 starts in the Premier League this season, but the England Under-21 international feels that he can return to his best form under Allardyce.
The 19-year-old is quoted by the Liverpool Echo as saying: "It's great to have his support. It has been a tough season for me so far but I have been getting back into the team and getting some games under my belt.
"The aim for me now is to try to cement myself in the team. We have got a lot of games still go and I am just looking to try and keep my place.
"My attributes seem to fit into what the manager wants from his midfield. With the way he wants us to play, with us pressing, it fits my game."
Davies will be hoping to feature when Everton host Crystal Palace at Goodison Park on Saturday afternoon.