Norwich edged closer to the Premier League trapdoor as Michael Keane's 55th-minute header was enough to secure Everton a 1-0 victory in sweltering heat at Carrow Road.
The Canaries remain six points adrift of safety with seven games remaining, while Everton move above Arsenal into 10th after edging an encounter low on quality in East Anglia.
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Daniel Farke made four changes ahead of their FA Cup quarter-final with Manchester United this weekend, but they had the better of the first-half chances as Onel Hernandez struck the outside of the post.
But Norwich were unable to respond to Keane's header from Lucas Digne's corner, leaving them in an increasingly precarious position at the foot of the Premier League table.
"We need an even bigger miracle now," Farke said afterwards. "But as long as we have a chance, we'll try to stay in this league. We will keep going and hopefully with this desire and commitment we can win points. So long as we have a chance, we'll keep going."
Norwich: Krul (6), Aarons (6), Lewis (6), Godfrey (6), Klose (5), McLean (6), Tettey (6), Duda (5), Rupp (5), Drmic (5), Hernandez (6).
Subs: Vrancic (5), Cantwell (n/a), Buendia (6), Pukki (6), Idah (5).
Everton: Pickford (6), Digne (7), Coleman (6), Keane (8), Holgate (7), Gomes (7), Davies (5), Iwobi (8), Bernard (7), Calvert-Lewin (6), Richarlison (5).
Subs: Baines (n/a), Sigurdsson (7), Gordon (n/a), Kean (6).
Man of the match: Michael Keane.
How problems mounted for Canaries
Those chances, however, are diminishing. No Premier League side with 21 points or less after 31 league games has survived - but Norwich carved out the only half chances of the opening period.
Farke opted to bench 11-goal leading scorer Teemu Pukki, and his replacement Hernandez saw a deflected shot hit the outside of the post after 18 minutes as Jordan Pickford scrambled across his line.
Farke claimed that hot weather could benefit his Norwich side with temperatures around 25 degrees at kick-off, and it was Everton who looked flat in the heat as Tom Davies was dispossessed in midfield on the stroke of half-time to allow Lukas Rupp to fire at goal from the edge of the box, but his effort was palmed to safety by Pickford.
Everton replaced Davies with Gylfi Sigurdsson at the interval and asserted their superior quality to take the lead within 10 minutes of the restart as Keane nipped in front of Timm Klose and Kenny McLean to glance home his second goal of the season.
- Teemu Pukki, Todd Cantwell, Emiliano Buendía and Tom Trybull all dropped to the Norwich bench as Alexander Tettey returned as captain alongside Onel Hernández, Ondrej Duda and Lukas Rupp.
- Everton made one change to the side that drew 0-0 with Liverpool in the Merseyside derby as Anthony Gordon dropped to the bench as Bernard started.
These are crucial days for the Canaries, beaten 3-0 at home to Southampton in their first game back since the lockdown, and it was Everton who nearly doubled their lead when Dominic Calvert-Lewin forced Tim Krul into tipping over his rising shot three minutes later.
Moments after the young English striker missed another fine chance to add to his 13 Premier League goals this term, glancing an Alex Iwobi cross wide, Farke introduced Pukki and Emi Buendia, and it was the Argentinian's free-kick that nearly resulted in a Norwich equaliser as Andre Gomes sliced his clearance just wide at the near post.
But Norwich are winless in their last 20 league games when they have conceded first, and that run was extended here as Everton saw out the final minutes to move four points off seventh spot and a potential Europa League place.
What the managers said
Norwich boss Daniel Farke: "Of course we're disappointed with the result today, but after the Southampton game I thought we were much better. A loss was not the fair outcome of the game.
"We showed during the whole game that we were much better defensively. We were solid, spirited, well structured and were the better side in the first half. We deserved something more, but one set-piece was the difference.
"Our defending was not switched on in that situation. We had six corners and were not able to score but in general I only have compliments for the lads for a disciplined performance full of desire that deserved much more.
"We know at this level we need to be outstanding to get points and have a little bit of luck. The set-piece was the difference but I can't ask any more from my players."
Everton boss Carlo Ancelotti: "I'm pleased as we didn't have the energy that we had against Liverpool. It's difficult to prepare for games every three days. We didn't play well and we were a little bit slow in the first half but we were much better in the second.
"It's important for the whole team to work towards the results, not just the defence. We've shown good spirit defensively and if you do this you can have a clean sheet.
"We need to have a goal, to have motivation, and this would be a fantastic achievement to reach - the Europa League. It will be difficult, the teams above us aren't too far but we have to run fast."
Man of the match - Michael Keane
Everton have been defensively organised since the restart, keeping a second consecutive clean sheet to consign the 4-0 thrashing at Chelsea to ancient history.
With Yerry Mina out injured, Michael Keane has grabbed his opportunity to be Mason Holgate's first-choice defensive partner and he capped a fine defensive performance with a trademark centre-back header from a corner.
Four of Keane's five goals in the Premier League have been scored away from home, including each of his three in the competition for current club Everton.
- Everton kept just their second away clean sheet in the Premier League this season, while this was their first away league win without conceding since March 2019 (2-0 against West Ham).
- Norwich suffered their 20th defeat of the Premier League season in this game - only once previously have they lost more after 31 games of a Football League campaign (22 in 1946-47 in the Third Division South).
- Since Carlo Ancelotti took charge, Everton have only lost one of their seven Premier League games against opponents starting the day in the bottom half (W4 D2) - 3-2 against Arsenal back in February.
Live Premier League
Everton take on Leicester at Goodison Park at 6pm on Wednesday July 1 live on Sky Sports Action (coverage starts at 5.30pm), while Norwich host Manchester United in the FA Cup quarter-finals on Saturday at 5.30pm before visiting Arsenal on Wednesday at 6pm.