Everton manager Carlo Ancelotti hopes Monday's 1-0 win over Southampton acts as a springboard in their bid to bring Champions League football to Goodison Park next season.
Richarlison scored for the third successive match inside the first 10 minutes to move Carlo Ancelotti's side within two points of fourth-placed West Ham with a match in hand.
But, while Everton's worst home run since 1958 was brought to an end, the pain for Saints continued with just one draw and eight defeats in their last nine league matches leaving them seven points above the bottom three and wondering just where their next win will come from.
Speaking afterwards to Sky Sports, Ancelotti said: "To fight for European places we need to improve the home run, this performance and victory can help us have a better run at home.
"It will be a dream to be in the top four at the end of the season. We are quite close, but the season is long - we have to stay in our goal which is to play Europa next season."
Everton can move into the Champions League places if they beat West Brom on Thursday, live on Sky Sports.
"We have a great opportunity to be in the top four and I think we'd like to touch the position," Ancelotti added later in his press conference.
It's an opportunity to be there. We can be in the top four. There will be a big fight until the end of the season and until the last game.
"We have to put all our efforts into the game on Thursday to try to be there. There is a real possibility to be in the top four in the next game, to see how we feel if we are there.
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"It's an opportunity to be there. If we win [against West Brom] we'll be in the top four and there will be a big fight until the end of the season.
"We're pleased that we can fight for that position but this doesn't change our target, which is to be in Europe next season."
Ancelotti believes consistency is the ultimate key to Champions League qualification but what he could really do with is Richarlison catching fire in the final three months of the season.
His angled ninth-minute finish was only his fifth in the league this season but, since the start of last season, Joao Pedro (30) and Paris Saint-Germain's Neymar (19) are the only Brazilians to score more goals in the top five European leagues than him.
Playing in a two up front with Dominic Calvert-Lewin, who has 13 league goals this season, the Brazilian took his chance brilliantly.
"Richarlison is not doing something extraordinary now," Ancelotti added. "He's doing his job and he's a player with this kind of quality. I hope and I'm sure that he's going to continue.
"Richarlison is a fantastic talent, we have two fantastic strikers with different characteristics.
"When you are 1-0 up the last minutes are always the same, the opposition team puts a lot of power in and we had to control. They had practically one opportunity. Pickford was good in the goal, really secure. I am really pleased, honestly."
'Top-four finish? Anything possible with Ancelotti'
Sky Sports' Jamie Carragher:
"I think Everton may just fall short of the top four - the squad of players is maybe just a bit short. But if they are to get into the top four, a lot of it will be down to the record of manager Carlo Ancelotti.
"Everton's last couple of wins have taken his win percentage to over 50 per cent, and no manager has been close to that since Howard Kendall's first spell.
"That is Everton's best-ever manager, so that just shows the job Ancelotti has done since joining the club. With Ancelotti in charge, anything is possible in terms of the top four.
"Jordan Pickford wins Everton two points with that late save. When you also think about his performance last week at Anfield. Everton did brilliantly to get their head on so many set-pieces. [Jannik] Vestergaard was slipping but it was a big moment for Pickford.
"He's a player who had the finger pointed at him, but he's been in great form the past two games, hence why Everton have kept clean sheets."
Hasenhuttl rues lack of cutting edge
Southampton's first real opportunity fell to Moussa Djenepo in the final 10 minutes - and he blazed wide when well placed - but it was an escape for Everton, who faded after having a second goal ruled out for offside midway through the first half.
Saints also came close to a late leveller when Pickford dived at the feet of Jannik Vestergaard, but the visitors slumped to an eighth league defeat in their last nine games.
Boss Ralph Hasenhuttl said: "It wasn't a good start with them scoring from their first chance. It was a long ball and we didn't defend it well. It was a goal from Richarlison with them running from deep in the same way he did against Liverpool and we couldn't defend it in the same.
"From Everton, they had a few chances from set-pieces but not a lot else. At the moment, we just can't score and this is our problem.
"We didn't have so many chances today, and in the last third, the last pass to score is not there at the moment. We're not strong enough. The last two goals we've scored were from Minamino but he was out today.
"In the second half we tried a lot. I brought three young lads in and they did what they could at the moment given the injuries we have at the moment. The good thing for me is that Stuart (Armstrong) showed me he can play in the six.
"In this position, Oriol (Romeu) is out for the rest of the season so it gives me a little bit of hope and it helps us.
"We have to work for the luck and at the moment it's not there. It was by far one of the best second-halves we've played in a while. Things have to be perfect for us to win games. There's not so much missing I think and this is what we have to tell the guys."
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Everton travel to face West Brom on Thursday, live on Sky Sports Premier League; kick-off is at 6pm.
Southampton visit Sheffield United on Saturday, also live on Sky Sports Premier League; kick-off is at 3pm.