A Richarlison header 57 seconds after half-time gave Everton a 1-0 win at Sheffield United to end their hopes of Europa League qualification.
The Blades' defeat left them hoping Crystal Palace would do them a favour in the race for Europe at Wolves later on Monday night, but the hosts' 2-0 win at Molineux ensured Chris Wilder's men can no longer catch the Black Country side.
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Everton came flying out of the blocks in the second period after an opening 45 minutes where both sides struggled to find their rhythm, and showed a touch of class to lead when Richarlison expertly aimed Gylfi Sigurdsson's free-kick beyond Dean Henderson.
Sheffield United rallied without managing to test Jordan Pickford while Henderson was alert to keep out a deflected Dominic Calvert-Lewin effort, as the Blades remained eighth in the Premier League, while Everton rose back above Southampton on goal difference and into 11th.
Everton show potential; Blades inconsistency goes on
With Everton's season over bar the shouting, it was understandable their intensity levels appeared lacking in the early stages at Bramall Lane, but the hosts' drab performance was more of a surprise.
The Blades pulled only a solitary save from Pickford before half-time - and that came from a Chris Basham cross - while Everton went close through an early Richarlison effort, and should have later found themselves in front before the break.
Sheffield United: Henderson (7), Baldock (6), O’Connell (6), Basham (7), Egan (6), Stevens (6), Norwood (6), Berge (6), Osborn (6), McBurnie (5), McGoldrick (5).
Subs: Fleck (7), Sharp (6), Lundstram (6), Zivkovic (n/a).
Everton: Pickford (7), Sidibe (7), Branthwaite (8), Keane (7), Digne (7), Davies (7), Gomes (7), Walcott (7), Sigurdsson (7), Richarlison (8), Calvert-Lewin (7).
Subs: Gordon (n/a), Coleman (n/a).
Man of the match: Richarlison
Theo Walcott returned the ball to the back post after a half-cleared corner where Dominic Calvert-Lewin should have found the back of the net against his boyhood club, but instead rattled the woodwork from a matter of yards.
Richarlison would let him off the hook less than 60 seconds after the teams returned to the pitch though, getting a firm header on Sigurdsson's free-kick from deep on the right flank to leave Henderson helpless.
That galvanised Everton's second-half performance, which could have improved on the scoreline had Henderson not got a strong hand to Calvert-Lewin's deflected effort.
Andre Gomes tried his luck from distance with time running out as the Toffees began to flex their muscles without causing the goalkeeper too many further problems, while Pickford remained a relative passenger in the Everton net throughout.
Sheffield United named an unchanged line-up despite losing at Leicester last time out on Thursday.
A Jarrad Branthwaite debut was one of four Everton alterations made by Carlo Ancelotti, with Djibril Sidibe also making his first start since the restart.
That gave Everton a welcome win after four games without, while Sheffield United watch on nervously to affairs at Molineux as they hope their outside shot at European qualification remains in tact, which were later ended by Wolves' victory.
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Sheffield United boss Chris Wilder: "Top six should never have been in the equation from the minute we set foot into the division. What you try and do in a season is try and get your players and club to overachieve and we have ridiculously overachieved.
"Immediately after the game I am disappointed for the players. we had a real go at it but they just lacked that bit of quality. Sometimes I shake my head when I see the players the opposition can bring on. We have overachieved and there are quite a lot of clubs that would want to swap places with us.
"I thought we deserved something from the game. We let ourselves down with the quality at the top end of the pitch."
Everton boss Carlo Ancelotti: "The spirit was really good. It is not a surprise. The players know really well what our supporters ask for on the pitch - to have a good spirit and fight and we showed that today.
"We had opportunities on the counter attack, I am really pleased with the performance. We are at the end of the season, but this fantastic spirit can be a good sign for the future.
"Since December we did quite well. We lost a bit of motivation for the last few games, but it is important to finish well. A win here is important."
Man of the Match - Richarlison
The Brazilian's brilliant header proved to be the difference with the goal capping another excellent display from the 23-year-old.
His pace and trickery was a constant threat to the Sheffield United back line and he came up with the crucial moment of the game to dent the Blades' European hopes.
On the goal, Ancelotti said: "It was a super goal. Richarlison is really good with his head and the delivery was really good, but the finish was fantastic."
Opta stats - Everton end barren run at Bramall Lane
- This was Everton's first away league win against Sheffield United since October 1972, ending a run of eight successive visits to Bramall Lane without victory.
- Sheffield United have lost four of their nine Premier League matches since the restart (W3 D2), as many as they had in their previous 21 before the break (W9 D8 L4).
- All five of Everton's away wins in the Premier League this season have been by a one-goal margin. They're the only side in the division this season to have had 100% of their away wins by a single goal.
- Sheffield United failed to register a shot on target in a home league game for the first time since August 2016, in a 3-0 defeat to Southend in League One.
With all teams playing their final game of the season on Sunday at 4pm, Sheffield United travel to Southampton to bring down the curtain on their campaign. Everton host Bournemouth in their 2019/20 swansong.