Allardyce, appointed as Everton's permanent new manager on Thursday, will not have been impressed by their dire first-half display, but Sigurdsson struck soon after the interval before Calvert-Lewin fired home a crucial second.
Tommy Smith drew a simple save from Pickford as David Wagner's men sought a route back into the game, but the day belonged to Allardyce and his Blues' charges as Everton secured back-to-back wins and clean sheets for the first time this season. Huddersfield, seeking their second away win of the season and their first victory at Everton for 80 years, nearly took the lead when Tom Ince's curled corner led to chaos in Everton's box before Laurent Depoitre was thwarted by Jordan Pickford. "It could have been better for entertainment, more passing and moving, but we can build on that as we go on", he said.
"I had some work to do tactically at half-time, mainly on Sigurdsson and [Aaron] Lennon not running too far back".
The Terriers, meanwhile, have seen their abject away form continue, with the loss to Everton their sixth in the last seven league trips away from Yorkshire. Kavanagh started officiating in the National League in 2012 and has worked his way up to the top flight, having seen over the 2017 League One Play-Off final between Millwall and Scunthorpe United.
He was not to be denied in the 73rd minute, though his goal had more than a hint of luck to it. "This was probably their first back-to-back wins this season".
"Let's hope it is onwards and upwards", Allardyce said.
And so, at least from a possession-centric point of view, this was ideal preparation for next weekend's derby.
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They're a solid unit with a coach who's been in charge for 11 years and who was an assistant to [Jurgen] Klinsmann before that. The city of Moscow alone outnumbers Iceland's entire population 40 times over, but this is a team of giant-killers.
"If I take a job I'm totally, fully, 100 per cent committed and I don't take the job in any other circumstances". "That gave us a platform".
England youth forward Calvert-Lewin, Everton's outstanding player, added a deserved goal when he ran on to Wayne Rooney's pass and scored via a deflection off Mathias Jorgensen. Heitinga missed the penalty, but Louis Saha got a fourth moments later, before Leon Osman completed the scoring with a header.
The sight of Lee getting so wound up in his Everton tracksuit reminded that these are unusual times at Goodison Park.
Improvements were made after the break, but Allardyce will have been left in no uncertain terms that Wednesday with the exception, rather than the rule. He was Allardyce's on-field lieutenant, enforcing positional discipline as he barked orders.
"I would like to focus on the players who are injured and get them fit as quickly as possible".
Once again deployed in a deeper role, the 32-year-old made a decisive impact on a Premier League match for the second time in less than a week. Such is the Allardyce ethos.