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
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
"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
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
Norwich boss : "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 : "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
"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 take on Leicester at Goodison Park at 6pm on Wednesday July 1 live on (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
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