Calvert-Lewin, who laid on the assist for Sigurdsson, capped off a good afternoon’s work 17 minutes from time after breaking clear and finding the net with the aid of a deflection.
It was an extremely encouraging performance for Allardyce, who saw his side graft in the opening half before moving through the gears to secure a second win - and clean sheet - in the space of four days.
The win elevates the Blues into the top half of the Premier League table for the first time since September.
At his official unveiling on Friday, Allardyce discussed his admiration of the performance turned in by the Toffees three days prior against West Ham United - he watched the 4-0 demolition from the Goodison directors’ box.
Therefore, it was no surprise to see the new boss stick with the same starting line-up. There were, however, returns to the bench for Oumar Niasse and Michael Keane, back available after suspension and injury respectively.
“All I’ve asked the players to do is repeat the performance of Wednesday and I think that will give us the best opportunity to try to win the game,” said the Everton manager in his pre-match interview. Fully understandable comments, considering the manner of the 4-0 victory and, having been appointed on Thursday, he had only had the opportunity to oversee one training session.
Allardyce was offered a warm welcome by all four sides of Goodison when his arrival in the home dugout was announced in the immediate moments prior to kick-off. He stood patrolling his technical area as a competitive but chance-less opening half an hour of football followed.
The tackles flew in and, indeed, Jonjoe Kenny and Tom Davies for Everton, and Scott Malone for Huddersfield, were penalised for late fouls with bookings by referee Chris Kavanagh.
The contest’s first shot came from the boot of the unlikely source of Cuco Martina. His curling right-foot effort from outside the area was competently pushed away by the Terriers’ Danish goalkeeper Jonas Lossl.
It was the best chance mustered by the Toffees in a half shaded on possession by the visitors from across the Pennines, although they too were quiet in attack, with a scramble inside the six-yard box shortly before half-time being Everton’s biggest concern.
The same 11 Toffees players reappeared after the interval but Kasey Palmer took the place of Danny Williams for Huddersfield.
Everton stepped it up right from the restart - and Sigurdsson struck the first goal of the Allardyce era with a clinical finish within 120 seconds of the second half commencing.
What a slick move, too. Aaron Lennon to Calvert-Lewin down the right channel, followed by a clever back-heel into the path of the Icelander, who celebrated his goal with a knee-slide in front of the Gwladys Street faithful.
Huddersfield tried to hit back straight away but Jordan Pickford had his angles covered as Tom Ince’s run-and-shot flew into the side-netting.
The visitors continued to push forward - the impressive Kenny crucially cleared inside his own six-yard area and Ince again tried his luck, only to see a wild volley land in the middle of the Park End.
No doubt the second period was much more eventful as the two sides traded chances. Calvert-Lewin stung the palms of Lossl and, from the resulting corner, Mason Holgate held his head in frustration, having ghosted in unmarked and glanced a header wide.
Morgan Schneiderlin and Ademola Lookman were brought on as the second half ticked by - and the game was put beyond the Terriers with 17 minutes left as Calvert-Lewin paced onto Wayne Rooney’s ball from deep, took a touch and found the net via a flick off defender Zanka.