The Frenchman struck a couple of times in the second half - the first of his goals just a couple of minutes after David de Gea had produced a top-class save to keep the hosts ahead - following on from Nemanja Matic's close-range finish 28 minutes in that put the Red Devils in front.
Three days on from thrashing Cardiff City 5-1 in Solskjaer's first match in charge, Pogba ensured that United just about remain in the hunt for a top-four finish, with this win lifting them to within five points of Chelsea and Arsenal, both of whom have a game in hand to play.
Huddersfield's losing run stretched to a fifth match at the weekend when going down 3-1 to Southampton, but they started brightly at Old Trafford and created the game's first opening.
A United side showing three changes from their first match under Solskjaer, including the enforced absence of illness victim Anthony Martial, took complete control of the first half from that point on.
A corner was flicked on at the near post by Victor Lindelof and Matic was waiting to convert from a few yards out - the Serbian midfielder's first United goal in 23 appearances since March.
Diogo Dalot should have joined his teammate on the scoresheet, only to volley wide following some impressive wing play from Marcus Rashford, though Town wasted a presentable opportunity of their own when Billing poked the ball into the side-netting instead of squaring it.
The change in personnel did not initially help United win back control of the match, as Town continued to press for a leveller and nearly found it through Depoitre's delicate flick towards the bottom corner, which De Gea brilliantly kept out to his right.
Pogba fired home a killer second goal for the hosts 64 minutes in, however, picking out the bottom corner with a fine finish once teed up by substitute Herrera, who profited from some good attacking play from Juan Mata inside the box.
Lingard then set up Pogba for his second goal of the afternoon - the Frenchman again placing the ball just out of Lossl's reach, though this time it was from 22 yards out when given too much time and space.
United were denied a third clean sheet of the season as Mathias Jorgensen pulled one back late on after getting on the end of Steve Mounie's flick-on, but it was Solskjaer who had the biggest smile at full time as the feel-good factor continues to build around Old Trafford a week on from Jose Mourinho's sacking.
MANCHESTER UNITED (4-2-3-1): De Gea; Lindelof, Jones, Dalot (Young 53'), Shaw; Matic, Fred (Herrera 53'); Mata (Gomes 80'), Pogba, Lingard; Rashford
HUDDERSFIELD TOWN (3-5-1-1): Lossl; Zanka, Schindler, Kongolo; Hadergjonaj, Kachunga (Lowe 57'), Billing, Mbenza (Quaner 62'), Durm; Pritchard; Depoitre (Mounie 73')