Amadou Onana and Michael Keane scored for the Merseyside club -- who were docked points last month for breaching profitability and sustainability rules (PSR) relating to losses -- giving them 16 points and 16th place in the standings, seven points clear of the drop zone.
Manager Sean Dyche told his Everton team, who would be sitting eighth if not for the deduction, to look at the setback as a challenge and they have risen to the occasion.
"I think it's definitely unified everyone at the club -- the players, the fans, the staff, everyone there," defender James Tarkowski told Sky Sports.
"Before the points deduction, we were playing well and winning games and so we've just continued that good work from that point, and we just keep building and keep picking up wins."
The 1.92-metre Onana headed home from the back post in the 19th minute, leaping high to send the ball past keeper James Trafford off a pinpoint cross from Dwight McNeill. "This is very, very important," Onana told the BBC. "The mentality of the team has changed and four wins in a row, very special. I am happy to be part of this team and hope it keeps going. We don't fear anybody."
Keane doubled the Toffees' lead six minutes later after Trafford saved his first attempt and he pounced on the rebound with a shot from the centre of the box to the bottom right corner.
Burnley, who have lost eight of their nine Premier League home games this season, are one spot above bottom side Sheffield United on goal difference.
"Bitterly disappointed," Burnley defender Dara O'Shea said. "We go into each game wanting to win and with the current situation we need the points. Being at home today we saw it as an opportunity. We conceded two poor set pieces.
"Disappointed but we have to look to next weekend now."
City drop points
Michael Olise scored a stoppage-time penalty as Crystal Palace clawed back from two goals down to draw 2-2 with Manchester City in the Premier League at the Etihad Stadium.
The draw kept City, who were without their league-leading scorer Erling Haaland due to a foot injury, in fourth place in the table on 34 points after a run of six games with only one win, and manager Pep Guardiola covered his face in frustration as the final whistle blew.
Palace are 15th with 17 points, eight above the drop zone.
"I think it was an unbelievable result," keeper Dean Henderson, who was playing his first league game for Palace, told the BBC.
"Manchester City are a world-class team and you've just got to try to stay in the game as long as you can and hopefully catch them on the break which we did."
In other matches, Newcastle United returned to winning ways with a 3-0 victory over 10-man Fulham.
And Chelsea scored twice in the second half to claim a laboured 2-0 win over Sheffield United.
- Reuters