Everton’s first set of three points in 112 days could be taken away from them this coming week but the impact of a most fortuitous victory may survive nonetheless. Sean Dyche’s team avoided equalling a club record 14 matches without a win thanks to a gift from the Burnley goalkeeper Arijanet Muric and pushed Vincent Kompany’s side closer to the Championship in the process. It was ugly by design, according to the Everton manager. No one knew whether he was joking. Everton’s taste of a first Premier League win since visiting Burnley in December owed plenty to luck and the visitors’ tendency to aim both barrels at their own feet. Dominic Calvert-Lewin deflected Muric’s delayed clearance into the Burnley net for a crucial match-winner while Dara O’Shea received a straight red card for a foul on Dwight McNeil, puncturing their chances of a recovery in an instant. Kompany’s strugglers still managed to make life fraught for Everton, who are expected to suffer a second points deduction of the season for breaching Premier League profitability and sustainability rules, but a defence guided superbly by Jarrad Branthwaite held on to a precious result. Dyche said: “Today’s win would put us on 35 points [but for the six-point deduction]. We only got 36 in a season’s work last season and we’ve got seven games to go. It’s difficult, we don’t know what is going to come next, but we can control the controllables. It is a very important victory and another step forward in the mentality to take the rest of the season on.” The importance of the relegation battle was lost on no one, with Goodison suitably tense, but the reason why both teams have toiled this season was also abundantly clear. The opening 46 minutes of play were a complete nonevent, bereft of quality, incident or any sense of urgency, until Everton fluked a breakthrough in stoppage time. It was a moment to forget for Muric. The Kosovo international dwelt too long over a routine clearance and gave Calvert-Lewin time to close in. The Everton striker stuck out his right leg in hope, Muric’s kick cannoned into it, and the ball looped over the keeper and sailed into the empty net. It was almost identical to Darwin Núñez’s opener for Liverpool against Sheffield United on Thursday. For Calvert-Lewin, who ended a 23-game run without a goal with his late penalty at Newcastle on Tuesday, it was evidence his luck in front of goal had finally turned. How Everton needed it. Branthwaite’s immaculate defending distinguished him from a very ordinary crowd. Dyche deployed a safety-conscious right wing with a combined age of 73 in Séamus Coleman and Ashley Young. André Gomes’s presence in central midfield offered hope of an improvement in Everton’s distribution but it was another hard watch as long punts sailed out of play and Burnley absorbed pressure with ease. “We looked at how well we had played this season and not won,” said Dyche, “so we deliberately tried to play it long and strong and play the game as awkward and ugly as possible and get an ugly win and it worked.” The visitors had produced the more enterprising moments before falling behind. David Fofana should have done better with a near-post header from a Josh Cullen corner but miscued his connection straight at Jordan Pickford. They suffered a second self-inflicted setback when O’Shea allowed Sander Berge’s pass to run away from him and upended McNeil as he pounced on the loose ball. The defender was the last man and the Everton winger was through on goal, but a long way from it when fouled. “There was a lot of force on the ball and it would have been an easy pick-up for the goalkeeper,” claimed Kompany. The Burnley manager added: “We frustrated the opponent and built momentum in the game but the momentum collapsed twice with moments of our own doing.” Muric atoned with a fine save from Calvert-Lewin as Everton produced a more aggressive, purposeful second-half display. The match-winner fired over after darting across the Burnley area while his replacement, Beto, was unable to capitalise on several counterattacks in the closing stages. Dyche argued Everton should have had a penalty for a foul on James Garner and Berge should have been sent off for fouling Beto when he was clean through. All that mattered, however, was victory. Everton finally had one.