Aaron Lennon darted in from the right to feed Calvert-Lewin and his neat back-heel fell into the path of Sigurdsson who calmly slotted home for his second goal of the season.
Allardyce was officially appointed on Thursday and the former England boss saw enough in the Toffees' performance to build upon after Gylfi Sigurdsson and Dominic Calvert-Lewin netted second half goals.
Huddersfield slumped to their fourth straight league defeat for the first time since November 2000 and after failing to score in eight successive games on the road head coach David Wagner admitted his players were low on confidence.
No doubt the new Everton manager and his coaching team will be keen to ensure it is something that continues, especially with a free-scoring Liverpool up next.
Sam Allardyce is no apologist; he will play his way during his Goodison Park tenure and the fans will simply have to accept it.
"I thought we were going to be better than that after Wednesday night".
Tom Ince went close a few moments later, running at Jonjoe Kenny before sending a left-footed drive into the side-netting.
It was the first his new side had managed in the league since an opening day win over Stoke City. "In possession, particularly in the first half, was a surprise". He and Everton have the feel of a marriage of convenience.
After Wagner's doomed attempt at playing with five defenders, he's likely to revert back to the more familiar 4-2-3-1 set-up that his players are fully acquainted with.
The Terriers' last win in this fixture came 61 years ago at home in Division One.
Faced with an energetic visiting side seeking to play on the front foot, the Blues' weaknesses were laid bare for the 40,000 inside Goodison and their new leader.
He was not to be denied in the 73rd minute, though his goal had more than a hint of luck to it.
Saturday's game will be the first for Sam Allardyce at Goodison Park.
Unlike the other promoted teams, Newcastle and Brighton, the Terriers have allowed 24 goals, the sixth most in the league.
Glenn Murray for the hosts pulled a goal back to threaten a fightback.
The Manchester-born official has never previously refereed Everton but was the man in the middle for Huddersfield's 0-0 draw with Burnley in September.
With T-Rex Tony (Pulis) also on the loose and seemingly sighted in the vicinity of a number of struggling Premier League teams, there seems to be no shortage of dinosaurs on the touchline and in boardrooms up and down the country for film-makers to cast their lenses upon. That gave us a platform to go and win the game.