Sheffield United boss Chris Wilder praised his players for "massive resilience" after the Blades' hopes of securing European football next season were dashed by Everton.
Richarlison's header at the start of the second half secured a victory for Toffees and means Wilder's team cannot finish more than eighth with a game remaining.
The Blades were favorites to make an immediate return to the Championship, but they exceeded expectations with an excellent season, in which they guarantee to finish in the top 10.
"The club and the players overcame en masse," Wilder told BBC Sport. "Having talked about jumping to the top 6 means that players deserve a huge amount of credit.
"We are delighted as a team that has made the most of the players. They left everything out there again."
"Yes, they didn’t have that quality in the third final, but between the top 10 guaranteed and another season in the Premier League – that’s what we should be thinking about."
Wolves win against Crystal Palace later on Monday, Sheffield United's little hopes of playing European football ended next season.
Richarlison's goal shortly after the break gave life to the game after a first half, in which Dominic Calvert-Lewin had the only real chance, but his header hit the post.
David McGoldrick was denied an immediate response to Sheffield United when his low shot inside the area was blocked, but the Blades seemed to be completely far from the brace.
Instead, it was Everton who seemed most likely to increase the advantage, as Calvert-Lewin defended Dean Henderson before Andre Gomes passed over the post in the final stages.
Everton's victory took them above Southampton and 11th.
"The spirit was very good," said Everton coach Carlo Ancelotti. "It is not a surprise, the players know very well what our fans ask for on the pitch – to have a good spirit and to fight, and we show that today."
"We had opportunities on the counterattack, I am very satisfied with the performance. We are at the end of the season, but that fantastic spirit can be a good sign for the future."
Blades' impressive campaign begins to take its toll
Wilder was extremely critical of his players after the 2-0 loss to Leicester last time, calling them League One and Championship, playing in the Premier League.
The 52-year-old is known for his loving approach to management – an attitude that has helped his team overcome the perceived individual ability this season.
Wilder clearly expected a positive reaction by naming the same side that started in Fox's defeat. His players faced a lot of huff and puff against Toffees, but they lacked quality in the third final.
There was also a distinct lack of urgency, with the work of a long but impressive first season in the Premier League, looking like it is starting to take its toll.
They pushed in the final minutes, when Everton leaned back, but ended the game without hitting the target.
Regardless, Sheffield United still have an excellent season game and can be proud of their efforts.
Richarlison underlines importance for Toffees
This was a much-needed victory for Everton, with the season in danger of failing.
A two-game winning streak and a draw after the restart were followed by a four-game losing streak that ended his hopes of finishing in the top 10.
Carlo Ancelotti approached this game with an eye on next season, evaluating his options for the club's youth team, offering promising 18-year-old defender Jarrad Branthwaite his first Premier League game, while 19-year-old striker Ellis Simms was included in the Bank .
But whatever changes Ancelotti decides to make to the squad over the summer, this game underlined Richarlison's importance in the Toffees' aspirations next season.
He was one of the few players on both sides who tried to make something happen in the first half with a few determined moves, while it was his goal – a superbly guided header after a smart run – that finally provided the spark that the game needed.
Richarlison has scored 15 goals so far this season – 14 of those in the Premier League – an impressive comeback in a tough campaign for Everton, in which they were at risk of getting involved in a relegation battle before Ancelotti's arrival in December.
A full preseason working with the Italian and the Brazilian has a real chance of evolving to 20 goals per season next year.
Game Man – Richarlison (Everton)
A rare blank for Blades – the statistics
- This was Everton's first away win against Sheffield United since October 1972, ending a series of eight successive visits to Bramall Lane without a win.
- Sheffield United have lost four of their nine Premier League games since the restart (V3 D2), as many as in the previous 21 before the break (V9 D8 L4).
- All five wins outside Everton in the Premier League this season have been by one goal margin. They are the only side of the division this season to have 100% of their away wins with a single goal.
- Sheffield United failed to score a goal shot in a home game for the first time since August 2016, in a 0-3 loss to Southend in League One.
- Everton have kept two goals in their last four Premier League games, having conceded 14 consecutive games on the road.
- Everton scored 14 header goals in the Premier League this season – only Liverpool scored the most (16). These are the most important goals of Toffees in a single campaign since 2010-11 (also 14)
- Richarlison has scored 26 Premier League goals since joining Everton – among Brazilian players in the top five European leagues, only Neymar (28) has more in that period.
- All three of Gylfi Sigurdsson's Premier League assists for Everton this season have been headed goals.
Sheffield United completes its Premier League campaign with a trip to Southampton on Sunday, July 26 (16:00 BST), while Everton host Bournemouth at the same time.