Saints missed the chance to close the gap on fourth-placed Manchester City to two points as they went down to a 2-1 defeat at the Britannia Stadium.
Schneiderlin opened the scoring with his first Barclays Premier League goal since September, but a mistake from Saints goalkeeper Kelvin Davis allowed Mame Diouf to equalise and substitute Charlie Adam stole all three points with six minutes to go.
Schneiderlin accepts the Europa League is a more realistic goal for Southampton now, but he insists the players still have faint hopes of qualifying for the premier club competition in Europe.
"We wanted to achieve the top four and now it's going to be even more complicated but we're not going to give up," the midfielder told Southampton's website.
"We wanted to give the fans the most prestigious competition but if not, we want to give them the Europa League
We want to secure European football for this club, that's our main target."
Stoke had struggled to create any decent chances before Davis' misjudgement handed them a way back into the game less than two minutes into the second half.
The 38-year-old declined to catch Steven Nzonzi's shot, believing it was going wide
The ball struck the angle and the goalkeeper had no time to respond before Diouf swept in from eight yards.
Saints manager Ronald Koeman was loathe to criticise his goalkeeper after the match.
"I call it unlucky," the Dutchman said.
"I will have to watch it back
I don't know if it was a cross or a shot, and maybe the wind was part of the difficulty for Kelvin."
Saints' demise was complete when substitute Adam nipped the ball away from Sadio Mane in the box and swept a volley past Davis.
After scoring a 65-yard wonder goal against Chelsea just a couple of weeks ago, Adam was unhappy at starting the game on the bench, but he thinks he gave his manager the perfect response.
"It's nice to keep scoring and win the game," the Scot said.
"It was disappointing to be on the bench and I think it was a great way to react by not starting the game."
The win moved Stoke to within four points of their 50-point target for the season.
And Adam believes the Potters can make up the one-point gap that separates them from eighth-placed Swansea.
He said: "We're edging nearer to doing better than last season and since the manager came in, our aim is to keep improving year upon year.
"We are a top-10 club and we want to consolidate ourselves in that position, so we want to keep pushing on for the remaining games of the season.
"It will be difficult, but we will keep pushing on and if we can finish in eighth place then we will be very happy."
Source : PA
Source: PA