Wayne Rooney wrapped up the points with a penalty – he had an earlier one saved before Calvert-Lewin knocked in the rebound – for his 10th goal of the season.

Victory sees Everton rise to ninth – a heady spot given they were in the relegation places on October 23 – and unbeaten in five matches under Allardyce.

“In games you need something special and Gylfi provided that,” Allardyce told Sky Sports. “From there it was nice and comfortable.

“It is a great three points considering how poorly we played in the first half.

“Top half, I think,” Allardyce said, when asked what his aims were for the rest of the season. “Let’s not get too carried away. Delivering 13 points from a possible 15 is an outstanding feat from where we came from.”

Rooney said he was not going to surrender penalty duties even though he has the second worst miss record in Premier League history – 10 from 33 with England and Newcastle great Alan Shearer missing 11 of 67.

“I’m a confident person and believe in myself,” Rooney told Sky Sports.

“If you look at the pens I have missed you will see I put a bit of top spin on the ball so someone could follow up! I’m not giving them up.”

He made the game safe for Everton by slamming home his second penalty of the night in the 73rd minute, with Lukasz Fabianski getting fingertips to the ball but not keeping it out this time.

“I was never going to give that penalty up,” Rooney said. “Frustrating missing the first one, so I put my foot through it.

“I thought tonight we were sloppy at times,” he said, “ … but we showed great character to come from a goal down.

“Four weeks ago, we would have lost that game.”

Swansea, who lost their sixth successive away match and have only one win in their last 10 games, are four points adrift of safety.

“The mood in the dressing room is very flat,” Swansea manager Paul Clement told the BBC.

“We’re bottom of the table on 12 points. We’ve got to focus on getting to 15 points at the halfway mark.”