Mauricio Pochettino's men have been this season's surprise package - currently sitting fifth, six points behind league leaders Arsenal - and missed out on going joint top after suffering a 2-0 defeat at the Emirates Stadium on Saturday.
That loss was only Southampton's second of the season, and Schneiderlin believes his side have what it takes to become a permanent fixture in the upper reaches of the top flight.
"In England I understand that people think the tradition is to be Chelsea, Manchester City, Arsenal, Spurs to be in the top four but we are here to change the tradition," he told The Daily Mirror.
"We want to upset the Premier League and be able to upset everyone. That is our job and what we want to do this year.
"What is realistic for us didn't change from last week. We want to get as high as possible. If that is being in the top three, so be it.
"We want to push as high as possible. We don't want to have any regrets because we know the way we play, the way we are doing we are going to cause teams many problems.
"We showed that we deserve to be there but if we want to stay there for as long as possible we need to correct those negatives like conceding two poor goals (against Arsenal).
"Because if you want to be in the top five or even more you can't allow yourself to concede those goals.
"We are looking forward to testing ourselves against teams like this and I think next week we will work hard on those negative things we did and try to correct them to be able to win against Chelsea and Manchester City."
Schneiderlin has scored one goal in 12 appearances for Southampton this season.
