The Saints have beaten Liverpool and Chelsea in their past two outings to move four points clear of the drop zone but midfielder Davis says they cannot rely on solely those triumphs to get them out of relegation trouble.
"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."