Jeff Hendrick's 72nd-minute strike heaped further pressure on Marco Silva as Burnley secured a 1-0 win over 10-man Everton at Turf Moor.
Seamus Coleman was sent off for a second bookable offence (56) to hand the Clarets the initiative, and Hendrick met Ashley Westwood's corner to volley home the winner as Everton's weakness at set-pieces was exploited.
Silva's Everton side threatened in the first half with Alex Iwobi guilty of missing their two best chances, but it is now one league point from four away league games, with the Toffees in 17th place, one point above the relegation zone.
Burnley are unbeaten in four league games and go from strength to strength, moving up to fifth in the table following a third home Premier League win of the season.
Burnley: Pope (6), Lowton (7), Pieters (6), Tarkowski (7), Mee (7), Hendrick (8), Westwood (8), Gudmundsson (7), McNeil (8), Barnes (6), Wood (6).
Subs: Taylor (6), Rodriguez (6), Lennon (n/a).
Everton: Pickford (6), Coleman (5), Digne (6), Keane (7), Mina (7), Schneiderlin (5), Delph (6), Sigurdsson (5), Iwobi (5), Richarlison (6), Calvert-Lewin (5).
Subs: Sidibe (6), Gomes (n/a), Kean (5).
Man of the match: Ashley Westwood.
How Burnley exploited Everton's weakness
Silva acknowledged the section of disgruntled travelling supporters who booed him at the final whistle, but he has called once more for togetherness heading into an international break full of introspection.
"It's important they keep the faith in us," the embattled Everton manager told Sky Sports. "The position in the table is not our image. We had a worse period last season but we reacted strongly. We believe in ourselves and now is the moment to do the same. West Ham at home is now a must-win game."
Silva, whose side have now lost four consecutive league games for the first time since January 2015, will hope to still be in charge by then, but his side started well in east Lancashire. Gylfi Sigurdsson had the game's first chance but Clarets 'keeper Nick Pope was on hand to tip his free-kick around the post after five minutes.
Sean Dyche made one change to the Burnley side that drew 2-2 at Aston Villa as Johann Berg Gudmundsson came in for the injured Jack Cork.
Marco Silva brought Alex Iwobi in for the absent Theo Walcott as Andre Gomes returned to the bench following his time out with a rib injury.
Burnley then very nearly capitalised on their opponent's persistence with zonal marking at corners as Hendrick met Johann Berg Gudmundsson's deep cross only to see his volley saved by the feet of Jordan Pickford.
Iwobi was next to threaten as he saw his shot deflect narrowly wide of the post via Ben Mee before he spurned a better opportunity when Matt Lowton produced a last-ditch block from close range.
Coleman was somewhat fortunate to receive only a yellow card after catching Erik Pieters on the ankle, but referee Graham Scott judged there to have been no malice as the half ended goalless.
Coleman's reprieve was short-lived, however, as the Everton captain was dismissed 11 minutes into the restart after catching Dwight McNeil in an aerial challenge, with referee Scott deeming it worthy of a second booking.
The hosts nearly made the extra man count immediately when Ashley Barnes met James Tarkowski's flick-on but blazed wide from the angle. But Burnley's corners were causing Pickford problems, and eventually they made one count.
Silva's men failed to learn their lesson from Hendrick's first-half chance, and the Republic of Ireland midfielder - who scored the winner against Everton at Goodison two seasons ago - was on hand to fire home unmarked from Westwood's deep delivery.
It was the 22nd goal Everton have conceded from a set-piece under Silva and, while Michael Keane missed a good chance to level, Burnley might have stretched their lead when Jay Rodriguez was denied by Pickford.
What the managers said
Burnley boss Sean Dyche: "It was a battle, and they're coming off a tricky run. We knew they would come here ready to play, and they had a massive result coming here last year (Everton won 5-1). But I thought the mentality was good from our players.
"It was tight, especially in the first half, and it was the same until the sending off. They went a bit more expansive, and with 10 men that's not so easy. We found the goal and then I thought we saw it out pretty well.
"We're not going to get carried away with being fourth. We know we can play better, but the players deserve a lot of credit. Consistency is the key."
Everton boss Marco Silva: "I think we did more than our opponent to win the game but it's difficult to play here and they are tough to play against. We were the better team in the first half but we were not clinical.
"The red card made a big impact and at the set-piece they were braver than us. We weren't clinical from our chances at set-pieces, and we needed to be braver. To gain confidence and to play at our best level, the results will help us.
"Unfortunately for us, it was a set-piece goal again. At that moment, we needed to be brave and until that moment we were really good but we had to react stronger. They blocked Morgan (Schneiderlin) in that moment but it's up to us to address it in the dressing room."
Analysis: 'Really worrying time for Everton'
Soccer Saturday's Matt Le Tissier...
"After eight games in the season, you wouldn't expect Everton to be sitting fourth from bottom in the table. Even before the sending off, they weren't dominating the match.
"They weren't creating too many clear-cut chances but they really weren't on top of their game. This is a really worrying time for Everton."
Analysis: Time running out for Silva
Sky Sports' Ben Grounds...
Everton's problems are staring Silva straight in the face, but his side show no signs of learning. A 22nd goal conceded from a defensive set-piece since the start of last season, and the sixth red card of Silva's tenure - both figures higher than any other club.
Leadership and discipline - two qualities that Burnley have in abundance under Dyche - were again absent among those in blue shirts. But chasing the game without their captain Coleman, Silva did not act like a man in desperate need of salvaging a point.
The Portuguese made like-for-like changes as Moise Kean replaced the ineffectual Dominic Calvert-Lewin, while Andre Gomes later came on for Morgan Schneiderlin, but Burnley only looked like adding to their lead.
To the bitter end, the 42-year-old is sticking to his principles, but this was yet another game in which Silva failed to rescue something from a losing position. Something has to change.
Man of the match: Ashley Westwood
Westwood took full advantage of Everton's insistence on zonal marking with Hendrick lashing in the midfielder's pinpoint delivery for the game's defining moment.
But it was the 29-year-old's overall performance that stood out, with the Burnley man regaining possession on 14 occasions - more than any other player, alongside his game-high two key passes.
How Burnley added to Everton's woes in stats
- Burnley are unbeaten in four Premier League games (W2 D2), after losing five of their seven before that (W1 D1).
- Everton have lost four consecutive Premier League games for the first time since January 2015.
- Jeff Hendrick has netted two goals in his last four Premier League appearances for Burnley, as many as he managed in his previous 30.
- 22 per cent of Burnley midfielder Jeff Hendrick's top-flight goals have come against Everton (2/9).
Following the international break, Burnley travel to face Leicester City in the Premier League on Saturday October 19 at 3pm, while Everton host West Ham earlier on the same day - kick-off is at 12.30pm.