Norwich City 0-1 Everton
Norwich's slim survival hopes were dealt a blow as Everton kept their European dream alive thanks to a Michael Keane header.
Daniel Farke's side went into the game six points adrift of 19th place at the bottom of the table, and needing to force a win to narrow the gap.
The four changes made by Farke frustrated Everton in the first half as the visitors were kept at bay, but Keane's nodded goal at the start of the second period gave the Toffees the lead.
The 27-year-old defender found himself unmarked and was able to turn a corner past Tim Krul in the Norwich net for his second Premier League goal of the season.
Carlo Ancelotti said before the game he believed that European football was a realistic target for his side and the win took them a step closer to reducing the gap, while Norwich's struggles in front of goal continued. The Canaries have now scored just once in their last seven matches.
"I can't accuse the lads at all today. After a 3-0 loss you're looking to be solid and defensive and that's what we were during the whole game. We were the better side in the first half and made the better chances," said Norwich manager Farke.
"I can just praise the workload, the togetherness. In terms of attitude, we showed today organisation and I can just give compliments. Everton worked many crosses and it was important today for the players on the wings to be competitive. We had good situations.
"The feeling (of the empty stadiums) is still a bit surreal. I got the feeling we were able to create a good spirit in the team. We don't give up, we have seven games and have a chance.
"It's not unrealistic for us to win the FA Cup or stay in the league.
"When you wear this yellow shirt you have responsibility and big pride. As long as there is a chance, we won't give up."
The game started evenly with both sides enjoying periods of possession, although Norwich were largely pinned back in their own half.
Everton almost looked to be caught out after Onel Hernandez weaved through the box before getting a shot away which deflected off the outside of Jordan Pickford's post.
But that was the closest the home side came to breaking the deadlock in the first half, although Lukas Rupp tested Pickford with a long-range effort.
Unlike their previous game against Southampton, where a bright start petered out, Norwich were able to keep their early momentum going into the second half - but Everton took control as the game wore on.
The Canaries were unable to turn their first-half chances into goals however, and were eventually punished for it when Keane was able to get above the defence and turn the ball past Tim Krul from a Lucas Digne corner.
Substitute Emiliano Buendia came close for Norwich after a quick one-two with Teemu Pukki - but the Argentinian was unable to get the shot away.
In one of the last chances, Dominic Calvert-Lewin took a shot but his long distance drive was straight at Krul.
Toffees boss Ancelotti said: "What we needed was three points, the three points arrived and now we can dream to climb."