The Saints beat Arsenal in the League Cup on Tuesday night to make it five successive victories in all competitions, despite losing a raft of key players over the summer.
"The atmosphere in the training room is something really special," ESPN quotes Forster as saying. "There is a real kind of togetherness.
"As soon as you walk in the door you're made to feel really welcome, that is key when you come to a new club, everyone helps you settle.
"There is that real togetherness and that eagerness for the players to perform and to help each other out. We've all settled really quickly. It's been a great start, but it's only a start and it's important we keep moving forward."
Forster joined Southampton from Celtic in a deal worth £10m in August, becoming the most expensive British goalkeeper of all time.