The Gunners' 2-0 win at Goodison Park leaves them 11 points adrift of leaders Leicester City with eight matches to play, having played one game fewer than Claudio Ranieri's side.
A run of three league games without a win appeared to have ended their chances of claiming a first top-flight crown since 2004, but Koscielny says they will treat every remaining match as a final to keep their hopes alive.
"We failed in some important games," the defender said, speaking to a media conference. "When we had to score, we did not, we put ourselves in a difficult situation.
"We had to look forward and we did it this weekend at Everton. There are eight games left, eight finals to hope for something."
Koscielny feels injury problems are to blame for derailing much of Arsenal's season after some promising form before the turn of the year.
"We had a correct first half of the season," he said. "We had a lot of injuries in February. When you have [Santi] Cazorla missing for four months, [Danny] Welbeck returning in February... it's hard to compete with some teams."
Koscielny also jumped to the defence of manager Arsene Wenger, who admitted to being troubled by seeing supporter banners calling for his departure in recent weeks.
"Arsene Wenger has been here a long time. I think he saw other banners, he is very calm, he knows what he brought to the club," he said.
"I can understand the frustration of the fans, we gave ourselves a chance of winning this trophy. I think the coach is still the man of the situation. Mathematically everything is still possible."
Despite seeing Arsenal's title chances grow slimmer, Koscielny was keen to praise the efforts of Leicester this season, and paid particular tribute to midfielder N'Golo Kante, who earned a first senior call-up to the France squad for this week's friendly matches.
"He's had a very good season," Koscielny added. "He shows his qualities every weekend. He's a very important player for Leicester who can recover a lot of balls and help attacking players in the forward phases.
"Leicester have had a very good season, nobody would have bet on them. Everyone thought they were going to stop at one stage or another.
"They are still there - they get the best from each player, they're a strong team who defend together, they often score first and after that they defend the goal and look to hit in attack with very good players like Riyad Mahrez or Jamie Vardy."