The Ecuador international came off the bench to enliven a dreary occasion, scoring the 73rd-minute opener and winning the late penalty which allowed Leighton Baines to double their lead before Romelu Lukaku gave the scoreline a flattering appearance.
That late show papered over some of the cracks which were evident in the preceding three-quarters of a game which seemed to be meandering towards a dull draw until Valencia’s introduction.
Valencia arrived on loan from West Ham on the final day of the summer transfer window but it was the reopening of it in January — with Ronald Koeman looking to considerably strengthen his squad over the next month — which sparked his best performance and a first goal after making just 12 appearances, most as a substitute.
“He can play a lot of different attacking positions and give the manager a lot of possibilities to change systems,” said Koeman.
“It is too early to make a final decision about his future but it was a good signing for us.
“He was one of the players who changed the game, scored the first goal and created difficulties for Southampton.”
Koeman had preferred 19-year-old Dominic Calvert-Lewin to support Romelu Lukaku, scorer of Everton’s third after the Belgium international allowed Leighton Baines to score the second from the penalty spot, in attack.
The experiment lasted just 12 minutes before the teenager’s first start for the club was cut short by injury.
“We had to make a decision whether to bring in the boy and make Enner wait,” said Koeman.
“I wasn’t happy with the support we gave to Rom from the midfield in the last few weeks so that was the reason we started Dominic behind Romelu.
“We need to wait until tomorrow (for the injury assessment). We are disappointed but he is young and he will come back.”
Koeman also backed Baines as Everton’s number one penalty-taker after the full-back initially appeared to be ready to hand over duties to Lukaku when the striker asked for the ball. “Leighton is a good penalty-taker. All his life he has scored penalties,” said Koeman.
“Everybody likes to help Romelu to be top scorer in the Premier League but it was 1-0 and Leighton is still first.
“If it is 2-0 in the last minute I can understand letting Lukaku take the penalty but it was still an important time.”
Southampton’s third straight defeat made for a miserable festive period for them and boss Claude Puel.
The scoreline was as harsh on Saints as much as it flattered Everton in a game which was low on quality.
“It is difficult to accept this. Three-nil is too much to take. My players did not deserve this,” said the Frenchman.
“It’s a shame, because I saw a good team, with courage, good spirit, and a good attitude.
“It was an even game for 70 minutes but the first goal changed the face of the game.
“We have played three games in five-and-a-half days. It is very difficult and it is not possible.
“We tried to give a good answer but it is difficult in this situation.”
Robles, Coleman, Williams, Funes Mori, Baines, Gana, Davies ,Lennon (Valencia 62), Calvert-Lewin (Mirallas 12), Barkley (Barry 85), Lukaku.
Forster, Cedric (Stephens 6), Fonte, Yoshida, McQueen, Clasie (Long 78), Romeu, Ward-Prowse, Tadic (Boufal 67), Rodriguez, Redmond.