The Saints suffered only their second loss of an impressive campaign at the Emirates Stadium, which was largely down to a blunder by goalkeeper Artur Boruc, who gifted Olivier Giroud an opening goal midway through the first half, with the France forward securing victory from the penalty spot on 86 minutes.
Southampton had plenty of possession, but found clear openings limited, as goalkeeper Wojciech Szczesny saved well from both England new boys Adam Lallana and Jay Rodriguez.
Schneiderlin insists he saw nothing to deflect belief from within the St Mary's club - which not so long ago was languishing in League One - that they can last the distance through to May.
"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 said.
"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.
"We all want to get as high as possible. If that is being in the top three, so it is.
"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."
Southampton travel to Chelsea next weekend and, after playing Aston Villa, will host Manchester City.
Schneiderlin insists Mauricio Pochettino's men will be up for those challenges, having already beaten Liverpool and taken a point at Manchester United.
"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 as soon as possible to be able to win against Chelsea and Manchester City," he said.
"It is not one defeat and we are going to put our head down.
"We know that we had a very good start to the season and we want to carry on."
Schneiderlin vowed the Southampton squad will rally round their Polish goalkeeper.
"Artur is a great keeper, very experienced guy and he is going to save us some more goals because he is a competitor," he said.
"I am sure that next week he is going to be spot on and going to have a great game in the upcoming games.
"He has been absolutely fantastic for us since the start of the season and made some great saves."
Schneiderlin added: "We just need to be a group. When someone makes a mistake we are all together in the same boat.
"Something will happen next time. It will be another guy who is going to make a mistake and Artur will be there to help him out, so that is football."