The 2-0 win over bottom side Sunderland extended their unbeaten Premier League run to nine matches, during which they have taken 21 points.
Although they remain in seventh, Arsenal, in fourth, are only six points ahead having played one match fewer and Koeman has urged his players to set themselves the target of catching them.
"The worst situation for a football player is you play for nothing," said the Dutchman.
"Yeah, (we want) the highest position but I like to play for something.
"Above us is still a gap but it is never ending, when it is five or six points it is still possible to reach them.
"But the first team behind us is West Brom (four points back) and we get West Brom at home in two weeks."
Crucial to Everton's hopes of European football, and at this stage that is more likely to be the Europa League, is striker Romelu Lukaku who scored his 17th league goal - making him the top flight's leading scorer - to wrap up victory after Idrissa Gana Gueye had smashed home his maiden Premier League goal just before half-time.
It could have been worse for Sunderland as the increasingly impressive teenager Tom Davies rattled the post with a volley but Lukaku's strike, a brilliant solo effort running from the halfway line, equalled the club record for Premier League goals of Duncan Ferguson, now on Koeman's coaching staff.
The Belgium international reached the 60-goal mark in 129 matches, 110 fewer than Ferguson who has played a key role in the 23-year-old's development.
"Rom went to Duncan and, of course, Duncan knows that Romelu will beat his record," added Koeman.
"I think he will be happy when the record is broken because that means Lukaku is scoring and, normally, if he scores goals it gives the team three points."
Lukaku's qualities were evident in the manner he scored, holding off former team-mate Bryan Oviedo in a sprint race with the ball at his feet before getting off a shot which took a deflection off the defender to beat goalkeeper Jordan Pickford.
"He's so strong and if he needs to make that run and it's a battle between him and the defender then he's really strong, he keeps calm in front of the goalkeeper and, maybe he was a little bit lucky, maybe there was a touch off the defender, but you know that type of opportunity he will score," Koeman said.
Defeat leaves Sunderland, who have taken five points from the same nine-match period as Everton, three points from safety.
Manager David Moyes, whose miserable run against his former club since leaving Goodison Park to initially take over at Manchester United was extended to five defeats, insists his side can generate the sort of momentum Everton currently have.
"We have got 12 games to go. I am looking forward to a good run," said the Scot.
"I am sure we have got one to come. A few teams have had theirs. We are not that far away.
"Sunderland have been in that position quite regularly."
