The Saints have won their last three matches to climb into second place in the Barclays Premier League table, with new manager Ronald Koeman making an instant impact at St Mary's after succeeding Mauricio Pochettino.
But they face a daunting Capital One Cup trip to Arsenal on Tuesday night, the Saints having won in that part of north London only twice in their history and not since an old First Division encounter at Highbury in November 1987.
However, speaking ahead of the third-round tie at the Emirates Stadium, Schneiderlin said: "Why can't Southampton win a cup this season? Anything is possible.
"We're going to go to Arsenal, play our game and do the best we can.
"The players at this club want to do something great in the cup, it's very important to us.
"It's the shortest way to get into the Europa League, which would be great for the club and for the fans."
Southampton are unbeaten in five games in all competitions since an opening-day loss at Liverpool over a month ago.
Schneiderlin admits he has been surprised by Southampton's best start to a top-flight for 26 years, since Chris Nicholl's reign as manager, and the France international said it is still a work in progress under Koeman.
"It's a surprise to have got off to the best ever start in the league after going through so many changes in the summer," Schneiderlin said.
"The coaching staff changed and a lot of players left, so of course it's great for us to be in such a good position after five games.
"But we had a very good start last year and it was difficult to get back up the league after going through a hard time in December
We're going to try to be stronger in that period.
"Every manager changes the style when they come into a football club.
"It's different to Pochettino, we don't press so high and so hard during the game and we are more controlled in defence.
"We're winning at the moment but we need to keep going and I know the gaffer wants to address some things."
Koeman fielded a strong side in the previous round of the Capital One Cup - a 2-0 win at Millwall - and the Dutchman plans to do the same again with Southampton taking 5,000 fans to the Emirates Stadium.
"It's great to have that number of fans behind the team," Koeman said.
"Okay, it's a difficult match but like I said before, in the League Cup against Millwall, we like to win everything.
"If a player's in good shape he's able to play.
"We don't take risks, but normally we like to play our strongest squad for Tuesday."
One possible change could see Sadio Mane make his debut after the Senegal striker was finally granted a work permit.
Source : PA
Source: PA