Ronald Koeman had an open goal against his former club but declined the chance to credit Everton’s late victory with the introduction of Enner Valencia. “Maybe the reason Everton were much better than Southampton in the last 30 minutes was because they had one less day to recover,” said the manager who switched from St Mary’s to Goodison Park last summer. Honesty was the best policy; Southampton were physically and mentally shattered by the time Romelu Lukaku brought their cheerless festive campaign to a close.

Valencia, the Ecuador international on a season-long loan from West Ham United, was a major factor in a third consecutive league defeat for Claude Puel’s side, opening the scoring 11 minutes after replacing Aaron Lennon and winning the penalty that enabled Leighton Baines to kill the contest shortly afterwards.

Lukaku’s vast second-half improvement also contributed to breaking the visitors and he sealed Everton’s victory with an emphatic finish in dying minutes, his 89th goal on his 200th appearance in English football. But there was no disguising the consequences of three games in six days on the visitors.

Southampton had the edge in the first half, Jay Rodriguez squandering a fine opportunity, and there was little between the teams until Koeman increased his attacking options with Valencia, and Lukaku thrived with the extra support as the weary, retreating visitors cracked.

“We started the season with many games,” Puel said. “We have played every three days, sometimes two, and we are 10th in the table with a programme like Real Madrid and Barcelona. Since Boxing Day we have played three games in five days and it is just not possible [three in six]. We tried to give a good answer, we did for 70 minutes, but we couldn’t give a good answer in the end. For a long time it was a balanced game. It is difficult to accept this result.”

Everton, who are close to signing the Charlton Athletic teenager Ademola Lookman in a deal worth around £11m, gratefully accepted the chance to stretch tired legs. Ross Barkley, the impressive 18-year-old Tom Davies, Séamus Coleman and, most of all, Lukaku probed continually late on, stayed patient, and punished a Southampton side that featured six changes to the team beaten by West Bromwich Albion on New Year’s Eve and lost Cédric Soares within seconds of the kick-off when he careered into advertising hoardings and suffered a head injury. Puel must have been resigned to a long afternoon from that moment on.

Koeman started with Davies and the 19-year-old Dominic Calvert-Lewin in an attempt to freshen his options – the first time two teenagers have started a Premier League game for anyone this season – but the striker’s full debut came to an unfortunate and abrupt halt when he suffered an ankle injury. The disruption and fatigue produced a sterile first half in which Southampton’s defensive organisation and intelligence on the counterattack held firm.

They should have led when Rodriguez collected Jordy Clasie’s ball with his back to goal, spun away from Davies, evaded Ashley Williams and, with only Joel Robles to beat, skied his shot high over the bar. A costly miss.

Unsurprisingly, Everton’s energy levels told as the game wore on, with Lukaku’s influence improving in tandem with his first touch. Coleman’s raids down the right offered an important outlet against a deep defence and they combined to telling effect for the breakthrough. Lukaku released the Irishman to the byline and also met the inviting cross that came back with a downward header. Fraser Forster saved with a foot, the rebound bounced off Lukaku’s thigh and dropped perfectly for Valencia to convert from close range his first club goal for 12 months.

Valencia was then up-ended by Maya Yoshida as he attempted to turn on a pass inside the penalty area. Southampton had no complaints as Baines, Everton’s captain for the day, sent their goalkeeper the wrong way from the spot. Everton’s final goal was reward for Lukaku’s second-half contribution and evidence of Southampton’s lethargy. José Fonte was easily dispossessed by Baines as he attempted to bring the ball out of defence and Davies pierced the visitors’ defence with a fine ball into the feet of the Belgium international. Forster had no chance as an unstoppable shot flew high into his net.

“We needed something different up front,” Koeman said on the introduction of Valencia. “That’s why we started with Dominic behind Romelu. I wasn’t happy with the support from the midfield for Romelu in recent weeks but then we lost Dominic and you need to change it.

“You either bring Enner straight in or wait until late in the game. He was one of the players who changed the game and created difficulties for Southampton. It’s good to have him, he can play different positions and gives me the possibility to change systems, but it’s still too early to make a final decision about his future.”