In contrast to Farhad Moshiri’s critical assessment of Everton’s league position in midweek, this was just good enough from Marco Silva’s side. Just good enough to beat Bournemouth, just good enough to claim a first win in five home league games and more than enough to spark an eruption of joy and relief from the Everton manager. Pressure eased.

Silva’s impassioned reaction to the stoppage-time goal that sealed victory said everything about the scrutiny that has been intensifying on his position in recent weeks, and not only since the majority shareholder’s comments at the club’s general meeting on Tuesday. Dominic Calvert-Lewin’s cool finish, added to Kurt Zouma’s first goal for the club, delivered three hard-earned and invaluable points as well as welcome respite for the Portuguese coach.

“It is a really important win,” the Everton manager said. “It is always important, of course, but I understand why you ask about this win because we didn’t achieve what we wanted as a team or a club in the last weeks.”

Their current position of 10th will not be cause for long-term celebration given Moshiri’s ambition and level of investment, but there was no overstating the importance of this victory. Eddie Howe’s visitors started and finished strongly but departed only with regret: that they did not capitalise on their strongest moments, that André Gomes escaped a red card and the concession of a penalty, and that an 11th defeat in 14 matches will eat away at their belief.

“When you don’t win confidence becomes an issue but you wouldn’t have known it from our performance today,” said Howe. “We created chances and didn’t take them. We switched off for a second at a set piece and were punished. That changed everything.”

Everton took 31 minutes to create a chance of note against a Bournemouth side missing the injured Callum Wilson, a rumoured target for Chelsea. Until that moment they posed no threat whatsoever to Asmir Begovic’s goal and appeared to have no clear idea of how to mount any. The visitors controlled the early exchanges, displaying greater urgency and purpose. Erratic distribution from Jordan Pickford and his defenders offered Bournemouth further encouragement for a first away win since Fulham in October. “They put high pressure on our build-up and we made too many easy mistakes in the first half,” said Silva.

Josh King had missed one excellent chance for the visitors before David Brooks, denied a strong penalty claim by the referee Anthony Taylor, was played through by Junior Stanislas after Zouma’s appeals for a foul went ignored. Brooks placed his shot beyond Pickford only to watch in disbelief as the ball struck the inside of the far post and fell for Gomes to clear.

Everton, their fans growing increasingly restless and their manager increasingly angry, began to match their opponents for aggression and precision in possession as the interval approached. Their performance was transformed as a result. Bernard headed their first chance wide from a Séamus Coleman cross and just failed to connect with a Lucas Digne delivery from the opposite flank. Michael Keane headed against the top of the crossbar after another inviting cross from the France full-back.

Dominic Calvert-Lewin celebrates with Ademola Lookman after making the result safe. Photograph: Andrew Yates/Reuters

The second half opened in a similar vein with the growing influence of Bernard, Gylfi Sigurdsson and Gomes ending Richarlison’s isolation as the lone striker. Richarlison went close to opening the scoring when, after his initial header had been blocked by Jefferson Lerma, his rebound was blocked on the line by the knee of Nathan Aké.

Digne was responsible for creating the breakthrough with another perfect cross after Lerma headed clear from a Sigurdsson corner. The full-back controlled beautifully on the left, taking him away from King and floated an inviting delivery to the near post, where Zouma powered a header into Begovic’s top right-hand corner.

Gomes, already harshly booked for dissent, was highly fortunate to escape a second yellow card for a foul on Steve Cook, another one on Stanislas and especially when he clipped Lerma around the head inside the penalty area late on. “I’m told it’s a stonewaller,” said Howe. Bournemouth applied sustained pressure in the search for a late equaliser but it was Everton who struck again thanks to the perseverance of Ademola Lookman.

The young winger regained possession from Adam Smith while grounded and was then sent clear down the left by Idrissa Gueye. Lookman sized up his options and pulled the ball back for Calvert-Lewin who side-footed inside the bottom corner via a slight deflection off Aké. Relief consumed Goodison – the technical area in particular.