Thursday's 3-0 defeat to Manchester City – their third loss in four league matches – left the Gunners 10 points adrift of fourth-placed Tottenham and represented a second beating in five days against City following Sunday's thrashing at Wembley in the Carabao Cup final.
Despite their latest disappointment, Ramsey refused to give up on qualifying for Europe's elite club competition and still harbours hopes of getting their hands on the Europa League trophy.
"No, [we're not giving up on the top four] because mathematically it's still possible," he told the club's official website .
"There’s 10 games left now so we have to really win every single one of them.
"That's our target. Hopefully we can finish in the top four with the Europa League trophy as well.
"Considering the situation that we’re in at the moment, that would be a very successful season for us.
"Things in football happen so quickly. There’s so much to play for, so hopefully we can win all our remaining games. It'll be tough but that's what we have to do."
The Gunners have lost to City three times this season, but Ramsey does not believe Pep Guardiola's side have become a bogey team.
He said: "We’ve had some good results against Manchester City at home over the past few years.
"This season, they seem to have kicked on again. It’s no coincidence that they’re top of the league. They’re playing some fantastic football and they have so much quality.
"It’s one of those things where we have to just go again to start to pick up momentum on Sunday.
Confidence is a clear issue for Arsenal, but manager Arsene Wenger is trying to remain focused on the next game, despite their recent form struggles.
"You go up by stairs but come down by the lift," an enigmatic Wenger said of his side's problems. "We go through a difficult patch, that's football.
"I don't compare it to any other situation I face before. I commit completely and stay committed and focus on the next game."