Roberto Martinez says it would be "too much of a blue sky" at present to contemplate Everton finishing in the top four.
The Toffees recorded their seventh Premier League win in a row on Saturday to rise above Arsenal in the race for Champions League qualification.
Last Sunday, Martinez's men turned on the style to beat the Gunners 3-0 at Goodison Park but they had to dig in to win 1-0 at relegation-threatened Sunderland.
"It was very, very hard," said Martinez to Sky Sports. "Obviously that's what we expected. That's why it's such a satisfying win from us, because in this league you have to be able to adapt from game to game.
"We came on the back of a really stylish and technical performance. Today was about greed, character, real concentration and then showing a little bit of quality in front of goal. We had to be at our best to keep this clean sheet from Tim Howard."
Martinez was delighted with the way his players stuck to their task.
"The way Sunderland have been playing this season, they've been really tough to break down," he added. "They've been well organised, and I think today it was very difficult to get that balance right - how you're going to get a good threat up front, and being solid enough at the back."
The game was eventually settled by Wes Brown's 75th-minute own goal, the ball deflecting in past Vito Mannone after Gerard Deulofeu played it across from the right.
"I felt we had clear-cut chances," continued Martinez. "Steven Naismith and Deulofeu... we created those openings, but overall we had that discipline throughout the game and knew what to do to get a result.
"This is as difficult a place as you're going to get in the situation that Sunderland are in, so it is very satisfying and shows the maturity of the group."
Martinez believes his players are handling the pressure superbly, with only five games now left to play.
"We're well aware of the difficulty of every game that you go into, when you get into the final third (of the season)," said the Spaniard.
"For one reason or another, everyone is fighting for its aims, for individual situations. As a team, we had a real focus and real concentration in everything we did.
"These are the games that really allow you to end up with a high points tally, it's not just about when everything goes well on the ball and everything you do is with style. I think today it was all the other aspects you need to control in a football game, to get a good clean sheet away from home and a good result."
When asked if he was thinking of a top-four finish, he added: "We don't look at it that way. I think it would be too much of a blue sky. All we want is to take it performance by performance.
"We still have a lot of competition for places, there's a really good edge about our play and now we're facing Crystal Palace on Wednesday and that's going to bring a lot of demands in the next seven days. It's a really exciting week, we're looking forward to getting back to Goodison and using the crowd in our favour and we need to recover as quickly as we can."
Leighton Baines added: "We've got some tough games coming up but we're confident at the moment.
"Everyone saw the quality of the performance last week, and the win we got there, but today was a different game and we showed we can do the other side of it as well. Although we played some good football at times in the first half, we knew in a game like this where Sunderland are fighting for their lives they were going to make it tough for us and we knew we'd have to dig in, and we did.
"Every game's massive now if you're competing for something. We now go into fourth place and it's ours to hold onto.
"We can play football, we can work hard, we're all in it together and we're ready to give it everything we've got. We understand there's a lot of work still to be done, but it's in our own hands."