Dutchman Koeman guided the Saints to a first Barclays Premier League win when coming from behind to beat West Ham 3-1 at Upton Park, where Schneiderlin scored twice.
It has been a testing start to life at St Mary's Stadium for Koeman, who saw the likes of Rickie Lambert, Adam Lallana, Dejan Lovren, Luke Shaw and Calum Chambers all follow former boss Mauricio Pochettino out of the south-coast club, who finished in eighth place.
However, with summer signings Graziano Pelle, the £8million Italian striker who scored his second goal in as many games after the midweek Capital One Cup win at Millwall, Serbian playmaker Dusan Tadic and goalkeeper Fraser Forster all starting to gel, Portuguese defender Fonte believes Southampton can quickly move forwards.
"For sure, we have lost some good players, but this team still has enough quality to have a strong season and with the additions the club were able to bring in, they have shown their potential," Fonte said.
"You know that if you come to this club, you have to play and not be afraid to have the ball, to come and get it, to be dynamic.
"We have been training on that - the manager keeps talking to us about it.
"The manager came with a few, but important details - how he wants us to play and what he wants us to do.
"We have shown that we can do it, but we need to be consistent."
Fonte added: "It is hard to say where we can finish, but we have he quality to be very competitive.
"If we keep working hard and playing like this we can give ourselves a good chance to finish if not in the same position then higher."
Schneiderlin has started all three league games so far, with Southampton insisting the midfielder is not for sale, having been strongly linked with a switch to join Pochettino at Tottenham.
On Saturday evening, Schneiderlin gave an interview with Southampton's internal media channel insisting he wanted to focus on his game, but stopped short of committing his future to the club.
Fonte hopes the 24-year-old will still be at Southampton when the transfer window closes at 11pm on Monday.
"We know what a good player Morgan is because we train with him every day," he said.
"Sometimes these things can play with your mind, but he has been a very good professional for us, and we are just glad that we still have him.
"Against West Ham, he has shown again why there is so much interest in him
We are pleased he is still here and hope he will keep being here."
Source : PA
Source: PA