Everton have lost just two of their last 17 games and need to maintain that form at Wolves on Saturday, where a win could put them level on points with Liverpool and Aston Villa who are in sixth and seventh places.
Moyes has continually played down Everton's chances of gate-crashing the European party after the team gave themselves a real handicap by taking only 14 points from their first 15 games.
But the outstanding performance and victory at fifth-placed Manchester City on Wednesday has again cranked up the levels of expectation at Goodison Park.
Moyes said: "We could come up on the blind side, but we need teams above us to falter, to really come apart. But my players are trying their hardest to achieve something despite the poor start we had to the campaign that really did for us.
"We struggled then, with injuries and form. But we have had our bad patch so we need to finish the season well, and see where it takes us.
"The players have to show their quality to finish higher up the table, and they are certainly going for it. But if we lose, our ambition would be over.
"The teams at the top are hanging in there, they are keeping themselves just away from us. But we won't be giving up even if it will be very tough."
Moyes has goalkeeper Tim Howard struggling with a thigh injury, but striker Victor Anichebe returns to the squad following a back problem. Midfielders Tim Cahill and Jack Rodwell should recover from minor hamstring problems.
Everton are seven points behind fourth-placed Spurs having played a game more, but Moyes admitted: "I'm not thinking fourth place is a possibility, all I am thinking is that we can try to catch any of the clubs above us.
"We have given ourselves a lot to do really at this stage. I am not being negative, just realistic.
"All we can do is win our next game and see where it takes us. But I am sure that Spurs, Manchester City, Liverpool and Aston Villa will be looking at the top four and hoping there is a falter somewhere.
"But they will not really be looking at us and where we are. Those teams and chasing fourth and will not see us as a real threat for that.
"We can get into Europe though, but words are cheap, we have to go out and do it and not make any mistakes ourselves."
Moyes has backed his players to show they are good enough, adding: "I feel our squad are as good as many in the Premier League, but it has taken some of them time to settle. Leighton Baines and Phil Jagielka fall into that category.
"We will just keep going and try to remain competitive. And these players have shown we can do that. This season we now have the momentum after a poor start.
"To achieve anything we know we cannot make even one mistake. A defeat now will kill us off, so we must be 100% focused until the end of the season."
Moyes has also played down speculation that teenager Rodwell could break into England's World Cup squad on current form, but still believes that midfielder Mikel Arteta could make a late play for a place in Spain's squad.
He said: "I don't think Jack will be involved in the World Cup this year. One day he will be a regular England international, undoubtedly. But it is too early, too soon.
"But I would expect that the England manager will already have his squad picked in his head by now.
"As for Mikel Arteta, if Spain are short of players in any way in their midfield and there's an opening, then Mikel is fresh and coming right back into form and scoring goals for us which is a real bonus"
But Moyes knows Wolves will make it tough for his side having produced a fine 3-1 win at fellow strugglers West Ham on Tuesday.
Moyes said: "Wolves will have great confidence after their recent run, which I believe has just about made them safe so the real threat of relegation has gone for them. That is unless they have a real disaster, which I doubt will happen to them now.
"Both of us have had big results this week, we did really well at Manchester City but for Wolves to win at West Ham has been huge for them, a really emotional moment for them which has probably saved them from going down."