Mauricio Pochettino's side have beaten Liverpool and Chelsea in their last two matches at St Mary's Stadium to help them move four points clear of the Premier League drop zone.
But Northern Ireland captain Davis says the club are still not out of trouble and must now focus their attention on their next game at Reading on Saturday.
"We are still so close and we won't get carried away," he said.
"We take a lot of confidence from the results we've got recently but we've got another massive game again next week to concentrate on.
"To be honest, I think we're safe enough in terms of the personalities in the squad.
"We all know what we have to go and give in each game to go and get a result and that's the way we are in the Premier League.
"We know what it takes to win games and we have to show that same desire in each and every game we've got."