Everton moved up to fifth in the Premier League as Richarlison's 65th-minute header was enough to secure a 1-0 win over West Brom at The Hawthorns.
Mbaye Diagne was denied a dramatic late equaliser in the third minute of stoppage time when his finish was ruled out for offside.
The striker saw an early header brilliantly kept out by Jordan Pickford before Dominic Calvert-Lewin went close as Sam Johnstone kept out his shot from close range.
But Everton took control in the second period and found the breakthrough when substitute Gylfi Sigurdsson's cross was turned in by Richarlison for his fourth goal in as many games.
The visitors' third straight league win moved them temporarily into the Champions League places but Chelsea's 1-0 victory at Liverpool later on Thursday squeezed them out to fifth. West Brom remain in 19th place after 27 games, nine points adrift of safety.
Inspired change keeps Everton on track
Everton were favourites to take their opportunity, but recent defeats to Newcastle and Fulham served as cautionary tales against sides in and around the battle at the bottom.
It has taken time, but West Brom are showing signs of improving under former Everton boss Sam Allardyce and went into the game on a three-game unbeaten run, having kept clean sheets against Burnley and Brighton in their previous two outings.
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The hosts made a bright start and were denied the opener inside two minutes when Conor Gallagher's cross was met by Diagne, but his header was brilliantly clawed away by Pickford.
West Brom had taken just four points from a possible 39 against sides currently in the top half of the table but they continued to impose themselves as Pickford was then forced into action again by Matt Phillips from distance.
Half of Everton's eight league defeats heading into the contest had come against teams currently in the bottom seven, but Carlo Ancelotti's side grew into the contest and came close themselves after 38 minutes.
A cute pass by Bernard inside to Abdoulaye Doucoure led to Dominic Calvert-Lewin setting up Richarlison for the shot but the ball skipped off Darnell Furlong's boot and flashed onto the roof of the net.
An even better chance was to fall to Calvert-Lewin before the interval as another Bernard pass fell kindly for the England striker after deflecting into his path off Ainsley Maitland-Niles but Johnstone kept out his effort from point-blank range.
West Brom began the second period in sharp fashion as Matheus Pereira's shot was dealt with by Pickford before another wicked Gallagher cross evaded Diagne at the far post. Ancelotti had seen enough and swiftly changed formation as Allan brought greater control to his midfield.
But it was his second alteration that paid instant dividends just after the hour mark. Sigurdsson had been on the field for 43 seconds when, with only his fourth touch, his cross was headed in by Richarlison after Diagne had failed to clear the Icelander's initial corner.
Sky Sports' Andy Hinchcliffe said: "What a substitution by Carlo Ancelotti, sending on Sigurdsson. That's what Everton have been crying out for: quality delivery into the box."
Everton nearly doubled their lead when Lucas Digne's cut-back was lashed over by Calvert-Lewin as Allardyce turned to Hal Robson-Kanu and Robert Snodgrass off his bench for the final 10 minutes.
Diagne thought he had snatched a point when he collected Maitland-Niles' cross on his chest before tucking his shot inside the post - but the offside flag came to Everton's rescue.
What the managers said
West Brom boss Sam Allardyce: "It is small moments that change the game. The first few moments with the header and Pickford saves it. Also what I consider to be a penalty for us for a foul on Diagne in the box. If we get 1-0 up, it is good for us. We start well and then slip up for one moment and concede.
"We score a great goal at the end and he is two millimetres offside and you realise you are not getting the luck you deserve. We now need to beat Newcastle on Sunday and get closer to the teams above us. I can give the lads a lot of credit but unfortunately no points.
"We have struggled to score goals from day one and that was the case again tonight. Everton had one lucky chance for Calvert-Lewin and the goal. We had more chances but we didn't take them.
"We were in the position to get more than one point today. We haven't even got that today. We need to correct that now on Sunday. It will be about mentality. Grit and determination and the size of your heart. We need to push through the pain barrier. We will still be feeling the mental and physical exhaustion of this game."
Everton boss Carlo Ancelotti: "I feel really good. If I look at the table, I'm really happy but there are two problems. First, I don't know how long it will be and unfortunately the season doesn't end today. We know that we will have to fight.
"At the start of the season, no one could say after 26 games Everton would be there in the top five so this is well deserved from my team as they're working really well."
"I talked to the players about following the dream of the Champions League. Most of these players haven't played one game in the Champions League. We'll have to fight and we'll need to sacrifice until the end."
Man of the match - Richarlison
There was relief for Carlo Ancelotti and his Everton side at the final whistle.
The visitors had been below their best, but they still had found a way of winning for a third straight game, bouncing above their local rivals Liverpool and into fifth spot.
It was a familiar source that proved the difference, as was the case against Southampton on Monday.
Sky Sports' Andy Hinchcliffe said: "Richarlison has underlined just how important he is to Everton. He's bang in form, and at times he's pretty much unplayable.
"He's got a bit of everything: great skill, pace, strength and is good in the air. Sigurdsson's ball in was an excellent one but he had to turn it home."
Player reaction: 'We knew we had to defend deep'
Everton midfielder Gylfi Sigurdsson told Sky Sports:
"Never in doubt! Keano (Michael Keane) assures me it was offside. Dominic (Calvert-Lewin) has scored a lot of goals but Richarlison is chipping in now at the right time. 1-0 in the last game and today.
"That is six points he has won us in the last two. It was a tough game. We knew they would defend deep and well. We had an idea a set-piece might win it. With Allan coming on he wanted us on the ball and playing. He did well and I got the assist.
"We are confident after a couple of good results. But it is tight at the top. We have to keep going and work hard. If we get on a run we have a chance."
Analysis: 'Everton must work harder for top-four dream'
Sky Sports' Jamie Redknapp:
"It wasn't a game full of quality but Gylfi Sigurdsson is a quality player. He has the ability with his right foot to put the ball into great areas. He doesn't try to over-hit the cross... it's floated in to allow someone to arrive onto it. It's a really good header from a standing jump by Richarlison."
Ashley Cole told Sky Sports:
"If I'm really honest and we wanted to break it down to Everton's challenge for the top four, they've got to work harder. I think off the ball, they've got to press with a little bit more intensity. I've seen players at 0-0 walking, and they weren't really at it.
"But if they want to keep fighting for the dream of the top four, Carlo will get them working harder."
Everton double up again - Opta stats
- This was Everton's fifth Premier League double against West Brom - against no side have they done so more in the competition's history.
- Everton have won nine Premier League away games this season, their joint-most on the road in a single campaign in the competition (also 2008-09).
- Everton have scored more headed goals than any other Premier League side this season, with Richarlison's winner their 11th such goal this term.
- No team has failed to score in more different home Premier League games this season than West Brom (8), with the Baggies also conceding a league-high 33 goals in home games this term.
What's next?
West Brom host Newcastle at The Hawthorns on Sunday; kick-off is at 12pm. Everton travel to face Chelsea at Stamford Bridge on Monday; kick-off is at 6pm.