The 26-year-old was signed from Celtic to take over the number one position at St Mary's and impressed on his home debut in Saturday's goalless draw with West Brom.
In a match where chances were at a premium, Forster made the save of the game as he kept out a late shot from Baggies' record signing Brown Ideye.
And the former Newcastle man is hoping to use such performances as a springboard to displace Manchester City's Joe Hart as Roy Hodgson's go-to goalkeeper.
"It was good to keep a clean sheet," he said.
"It was nice to make the save, it is one of those where I haven't really had that much to do save-wise and it is just important to try and help the lads out as they defended brilliantly in front of me
It is my job when one comes along to make those saves.
"I was delighted to have the interest from Southampton
They were brilliant last season, a few have gone out but we have got a lot of good players in.
"It is one of them where I have only played two games in the Premier League and it is a fantastic league to be a part of.
"It comes down to playing well for Southampton first and foremost - that is what it is all about
If I do all that then we will just have to see what happens with England.
"For me personally it is about me working as hard as I can with Dave (Watson, Southampton and England goalkeeping coach) and in the gym and improve myself as a 'keeper."
Koeman is still without a win after his first two Barclays Premier League games in charge, with his new-look side unable to break down a resolute West Brom defence.
James Ward-Prowse came closest when he clipped a post with a free-kick with Forster's late save vindicating Koeman's decision to replace Artur Boruc.
The Dutchman said: "The goalkeeper is a very good one, we know that because that is the reason that we sign players
He's 26, and that is a young age for a 'keeper.
"We knew the situation with Boruc, his contract and the fact that he is older and we have to change that
Forster brings qualities and experience and he showed that."
Saints fans lauded Forster with calls of 'England's number one' and Koeman believes that is not something beyond his capabilities.
"I can't say that because I'm not the coach of England," he added.
"I'm the coach of Southampton and I'm happy that he is playing for Southampton
Who plays in the national team is not my job.
"Of course he is capable, he was one of the three goalkeepers in the World Cup
If you take three 'keepers to the World Cup, all the three can play because they are all good.
"We like that he will be number one for England, of course, because that is good for himself, good for his experience and finally good for Southampton."
The other goalkeeper on display in the dreary deadlock was Hodgson's other regular squad member Ben Foster.
The 31-year-old started England's final World Cup group game against Costa Rica and West Brom head coach Alan Irvine is pleased to have his own international-class keeper.
"I'm delighted with the one I have got, absolutely delighted," he said.
"He is a terrific goalkeeper and really big personality, and is really important in our dressing room and I would not swap him.
"(He has a) great personality, great professional and he is able to get other people going as well and full of enthusiasm and full of energy."
At the other end of the pitch, Irvine was pleased with the impact debutant Ideye made after coming off the bench.
"It is early days for him," he said.
"He is a good finisher and if he gets into positions where he gets chances then he will score goals
It is a good start
That is really encouraging - there is more to come from him."
Source : PA
Source: PA