A 3-2 win over Swansea moved the Toffees to within eight points of fourth-placed Arsenal with a match in hand, and the Gunners still to come to Goodison Park.
While the Europa League remains the most likely outcome, Distin said they will continue to push for fourth place until it is taken out of their reach.
"The dream is always alive as long as it is mathematically possible," he told Press Association Sport.
"We are not in a position where we have to focus on the team above us. We cannot depend on them (dropping points) so there is no point looking at it now.
"Every single game is important. This one is no more important than the previous one. We just want to win games.
"You have to win. If you want to be successful, you can't start a game thinking it will be okay to draw this one."
Everton head to Newcastle on Tuesday looking to maintain their European challenge, but Distin admits they have to tighten up in certain areas if they are to finish the season strongly.
Having got themselves into a comfortable 3-1 lead thanks to Leighton Baines' penalty - cancelled out by Wilfried Bony's close-range strike in the first half - and goals from Romelu Lukaku and Ross Barkley early in the second half, they conceded a late Ashley Williams header.
In truth they never looked entirely comfortable closing out the game in the last 20 minutes of the game.
"We don't score many goals and we made it difficult for ourselves," the Frenchman added.
"At the moment we need to change things from set-pieces because we keep conceding from them.
"We will work on that but the late goal will not affect us mentally or our next performance."
Swansea have worries near the foot of the table as they have not won since Garry Monk's first match in interim charge in early February after the dismissal of Michael Laudrup, and have picked up just five points from six games to leave them just four points above the relegation zone.
Defender Ben Davies accepts they have a hugely significant few days ahead days ahead as they head to Arsenal in midweek before hosting fellow strugglers Norwich on Saturday.
"It is a massive week for us and we definitely need something out of the next two games," he said.
"We cannot afford to let our confidence slip and get into negative ways with everyone feeling sorry for themselves because then we don't have any chance of winning games.
"It is important we keep that confidence up and play the way we know we can. We've got to take it into our own hands."
Monk has failed to have any impact since being placed in temporary charge and he knows with no previous managerial experience he is likely to come under more scrutiny as the season comes to a conclusion.
"I'll be judged on results - that's the way it is and until I'm told any different I'll continue to work as hard as possible," he said.
"I am happy to be judged - that is the way it is."