Ollie Watkins' second-half strike gave Aston Villa a hard-fought 1-0 win at Goodison Park, as David Moyes suffered a defeat on his return as Everton manager.
Dominic Calvert-Lewin created a huge chance in stoppage time and the home side's hopes of a dramatic equalizer were dashed, with Everton having won just once in their last 12 league games.
Watkins missed a great chance in the first half but made no mistake, latching on to a pass from Morgan Rogers and sliding his shot past Jordan Pickford to score. His ninth league goal of the season.
The result ignited Villa's push for a Champions League spot, which has seen them move up to seventh in the Premier League, three points behind fourth-placed Newcastle. Meanwhile, Everton remain in 16th place, just one point outside the relegation zone.
Moyes told TNT Sports: "I don't think there was a lot of excitement in the game. I have no illusions about the job I have to do. We couldn't score again."
“We did have good control of the game, we did the right thing, a little bit of mistakes and a little bit of misalignment cost us the game.
"The position we are in is not as good as we would have liked, that's obvious. Sean (Dyche) has had some great results at the start of the season but we have to show more quality on the ball and that's "It's disappointing and I've told the players that.
"The character was never in doubt, it always shone and they tried their best to get a result but just couldn't get a result."
Back in 2002, when Moyes first walked down the corridors of the tunnel and into the dugout at Goodison Park, Everton were celebrating becoming the first English club to reach 100 seasons in the top flight.
Now, as he begins his second round, his first test comes in the most capped top flight of English football between two of the most successful founder members of the world's first football league.
After a record 214th meeting, Moyes knows that in order to extend that proud record into next season he must reinvigorate an aging squad and one that has become stale under his predecessor.
Calvert-Lewin, whose contract expires in six months, has been urged to step up and make his second appearance as a striker, who has not scored since the game against Villa Park in September. The ball passed.
He had two good chances to break the deadlock in the first half, his second shot being deflected off the line by the excellent Boubacar Kamara. It never felt like I was going in.
His final chance summed up his struggles. The pass thrown to him by Jesper Lindstrom wasn't perfect as the ball swung his way, but he should have hit the target. He has failed to score in his last 16 games, playing a staggering 1,212 minutes.
Abdoulaye Doucoure's shot was also easily saved from Vitalii Mykolenko's cross. There was plenty of effort but few chances after Watkins' decisive goal and ultimately there wasn't enough quality to deny Villa their first away league win since October and first since last April. A clean sheet away from home.
Moyes added: "At the end of the day, it's not just this game tonight but the other games as well and I think everyone can see that," he said. I don't just come in and say 'we're not good enough at getting things done'.
"By the way, I can't 'magically' change this, I just can't. We need to improve these levels. The club's current situation may not return to its original level." The level of market participants may have an impact.
"We do need to add to the squad if we can but we have to be careful not to do too much (due to profitability and sustainability rules) but we will definitely try and see what we can do."
Everton have failed to score in nine of their last 11 league games, scoring just once in that time. They have gone more than five hours without a goal, but Villa have their own problems to sort out, having lost five league games away from home, conceding at least twice each time.
For Watkins, it was a night of redemption. The striker had a clear chance to shoot in the first half after receiving a poor back pass from Ashley Young, but his shot went wide of the post.
Rodgers had already made a fine save from Pickford by this point, but England's No. 1 showed indecision in a crucial moment six minutes after the restart.
The Blue Jackets missed several 50-50 challenges up front before Jarrad Branthwaite's errant pass allowed Rodgers to break through and pass to teammate Watkins, with the Everton goalkeeper unsure of his position.
It was the fourth time this season that Watkins combined with Rodgers to score, this time enough to seal all three points, but Moyes' return failed.
In fact, Villa didn't contribute much in the second half other than scoring. The opener could have come sooner as Jacob Ramsey's shot went inches wide on the stroke of half-time. The visitors resumed good game management after that, with Ramsey blocking Branthwaite's shot from a half-cleared corner and the friction on the green working in Unai Emery's side's favour.
Considering Everton's offensive problems, Watkins' goals will always be enough.
"I'm very happy," Emery said. "It was a difficult game. We played 90 minutes and performed well. In the first half we had some clear chances and in the second half we tried to push and control more but we couldn't do it but we scored ”
"This win is very important. We have started the year with wins at home, in the Premier League and in the FA Cup. Away, we are in a bad situation and we have to break it."