Match ends, Stoke City 1, Everton 2.
Stoke manager Paul Lambert says his side are "well in the running" to avoid relegation despite defeat to Everton.
Cenk Tosun scored twice to help the Toffees to their first away win since December with victory over 10-man Stoke.
The Potters - whose goal was scored by Eric Maxim Choupo-Moting - have won just one of their last 12 Premier League games.
"The games coming up are probably the biggest games the club has faced in many years," Lambert said.
"It's not against us. We are well in the running. We are going to fight for everything. We will do everything we can to stay in this league."
Tosun opened the scoring for the Toffees with his third goal in as many games before Choupo-Moting equalised eight minutes later.
But, in a match plagued by snowy blizzards, the Turkey international added his second with seven minutes remaining with a diving header.
Stoke had Charlie Adam sent off in the first half for a lunging tackle on Wayne Rooney.
There was confusion when the card was handed out as referee Martin Atkinson incorrectly showed it to Xherdan Shaqiri before realising his mistake and dismissing Adam.
Stoke made five changes to their line-up and started well in snowy conditions. Adam had the first chance to score when he received the ball just outside the area, chesting it down before easing past Tom Davies and unleashing a shot that glided over the bar.
Davies himself could - and perhaps should - have scored just minutes later for Everton, heading a Wayne Rooney free-kick over Jack Butland's goal.
However, in a first half of sporadic quality, Adam's sending off provided the most dramatic moment - the 32-year-old punished after challenging Rooney with both feet off the ground.
Despair for Stoke
It was another disappointing result for Lambert's men as they extended their run of league games without a win to seven, meaning their decade-long stay in the Premier League looks increasingly perilous with just seven matches remaining.
Crystal Palace's away win at Huddersfield makes the Potters' survival task ever more difficult too, as the Eagles rose out of the relegation zone and left Stoke four points from safety.
With Arsenal, Tottenham and Liverpool among their opponents next month after the international break, Stoke face a real battle to remain in the top-flight.
Their result against Everton doesn't tell the full picture of the 90 minutes, though.
They will be frustrated to come away with no points after a performance in which they dominated large spells of the action.
Adam's dismissal boosted Everton numerically, but Stoke's performance levels were not obviously affected.
They were, however, left to rue Peter Crouch's header over the bar under pressure from Phil Jagielka in the first half.
Had the former England striker converted, Crouch would have become his side's all-time leading Premier League scorer and, potentially, given his side the opportunity to take away something from a game in which they impressed for large parts.
Tosun celebrates brace... and eyes Europe
Tosun's Everton career was slow to get going but with four goals in his last three games, the 26-year-old has finally found his feet in the Premier League following a £27m move in January.
"I think I've settled in very well, I feel better," he said after the game.
Tosun broke the deadlock with an easy finish from close range, but it was his second, 15 minutes later, which was the more stylish finish, diving low to steer the ball past Butland.
Following the up and down rollercoaster that was Everton's form during the early stages of 2018, things finally look to be heading the right way for Sam Allardyce's side as they recorded their first back-to-back wins since December.
Confidence will be starting to rise on Merseyside with the Toffees now comfortably in the top half of the table - enough, most would expect, to keep them out of the relegation battle that continues below them.
Now 12 points above the bottom three, fans will certainly see Tosun's arrival at Goodison Park as having played a large part in their climb to safety, yet he has got even bigger ambitions for the rest of the season.
"We want to win as much as possible and we still have the chance to play in Europe next season," he said.
'That's why we bought him' - what the managers said
Everton manager Sam Allardyce: "Under the circumstances we are extremely pleased. Considering the elements both teams competed with today they've given everything - even Stoke after they went down to 10 men.
"It looked a bit iffy for us again but Cenk popped up with a super goal and that's why we bought him. It's paid dividends for us and it puts us on the 40-point mark, and even though mathematically we are not safe I don't think anyone can catch us now and we can start to prepare for next season.
"I think the quality of the finish caught everyone out, including the goalkeeper, but it was very difficult today to put any quality together because of the wind and snow. Wayne Rooney kept us going in midfield and kept us prodding but breaking down 10 men can be difficult.
"There weren't that many chances and any chance can come in today's weather - anything could happen, likely a mistake in these conditions and luckily we were the ones that capitalised on that today."
Stoke manager Paul Lambert: "I couldn't ask more for the effort. We had 10 men for most of the game and got ourselves back into it. It's a disappointing result but effort, I couldn't have asked for anymore.
"The referee's position is probably a better angle than anyone has [for Charlie Adam's red card] and looking at it now, I can see why he has given it.
"The first Everton goal was offside and how he's not seen that I don't know. These are big moments and when you're fighting for your life you need those decisions to go for you.
"The weather is part of the game. There's no excuses, that's just the British weather I'm afraid."
- Everton have won just their second Premier League away game of the season, ending a run of five straight defeats on the road.
- This is the first time Stoke have lost both league meetings against Everton in a single season since their first campaign in the Premier League (2008-09).
- Charlie Adam was the first player to be sent off for a foul on Wayne Rooney in the Premier League since Sam Sodje for Reading in December 2006.
- Since 2008-09, Stoke have had 14 red cards in Premier League home games - only Manchester City (15) have had more in that time.
- Paul Lambert's side are winless in seven Premier League games - only West Brom are on a current longer run (8).
- Cenk Tosun has now scored in each of his last three league games for Everton, after failing to score in his first four for the club since joining from Besiktas.
- Tosun also became the first Turkish player to score a Premier League brace since Tuncay Sanli for Middlesbrough against Aston Villa in November 2008.
As the international break gets under way, Allardyce has plenty of time to prepare for Manchester City's visit to Goodison Park in two weeks' time, before the Merseyside Derby kicks off the following week.
For Stoke, it's a trip to the capital up next as they take on Arsenal on 1 April, before London travels north for Tottenham's visit six days later.
- 17ShawcrossBooked at 55mins
- 27B Ndiaye
- 16AdamBooked at 30mins
- 22ShaqiriSubstituted forBerahinoat 76'minutes
- 32SobhiSubstituted forChoupo-Motingat 72'minutesSubstituted forRodríguez Ruizat 79'minutes
- 11Rodríguez Ruiz
- 15Martins Indi
- 6JagielkaBooked at 82mins
- 10RooneySubstituted forSchneiderlinat 87'minutes
- 26DaviesSubstituted forCalvert-Lewinat 53'minutes
- 7BolasieSubstituted forHolgateat 90+2'minutes
- Martin Atkinson
- Shots on Target
Second Half ends, Stoke City 1, Everton 2.
Corner, Everton. Conceded by Kurt Zouma.
Substitution, Everton. Mason Holgate replaces Yannick Bolasie because of an injury.
Phil Jagielka (Everton) wins a free kick in the defensive half.
Foul by Saido Berahino (Stoke City).
Attempt missed. Theo Walcott (Everton) right footed shot from the right side of the box is close, but misses to the right. Assisted by Idrissa Gueye.
Foul by Michael Keane (Everton).
Saido Berahino (Stoke City) wins a free kick in the attacking half.
Substitution, Everton. Morgan Schneiderlin replaces Wayne Rooney.
Foul by Dominic Calvert-Lewin (Everton).
Joe Allen (Stoke City) wins a free kick in the defensive half.
Attempt saved. Dominic Calvert-Lewin (Everton) right footed shot from the left side of the six yard box is saved in the centre of the goal. Assisted by Leighton Baines.
Goal! Stoke City 1, Everton 2. Cenk Tosun (Everton) header from the centre of the box to the bottom right corner. Assisted by Theo Walcott with a cross.
Attempt blocked. Theo Walcott (Everton) right footed shot from the centre of the box is blocked. Assisted by Cenk Tosun with a cross.
Phil Jagielka (Everton) is shown the yellow card for a bad foul.
Foul by Phil Jagielka (Everton).
Badou Ndiaye (Stoke City) wins a free kick in the attacking half.
Dominic Calvert-Lewin (Everton) wins a free kick in the defensive half.
Foul by Glen Johnson (Stoke City).