Jose Mourinho caused a stir by dropping Paul Pogba - who was hauled off in the 2-0 loss at Spurs - for Huddersfield's visit and, while his absence was notable in midfield, United rarely looked like dropping points at Old Trafford.
Like Pogba, Mooy also had to settle for a place on the bench, although the Socceroos midfielder entered the action after just 33 minutes in place of Philip Billing.
The hosts were dominant throughout the first half and should have been given a penalty when Scott McTominay – Pogba's replacement – was barged into by Terence Kongolo.
Clear-cut chances were something of a rarity before the break, however, as United's play suffered because of a lack of width caused by Jesse Lingard, Juan Mata and Sanchez – making his home debut – all wanting to occupy central positions.
When United did alter their system in order to exploit the flanks, the breakthrough eventually arrived – Mata crossing to Romelu Lukaku for a neat, near-post finish in the 55th minute.
Sanchez then wrapped things up 22 minutes from time, tucking home after his initial penalty was saved to get off the mark for United, who cut Manchester City's emphatic lead to 13 points after the latter's 1-1 draw with Burnley.
After a minute's silence to mark the upcoming 60th anniversary of the Munich Air Disaster on Tuesday, United started brightly, creating two chances inside 10 minutes.
Lingard crafted the first, darting through the middle and tearing past two defenders before shooting left-footed at Jonas Lossl from the edge of the area.
And Chris Smalling wasted the second, heading into Lossl's arms with the goal at his mercy.
United were unlucky not to win a penalty in the 20th minute as Kongolo clattered into McTominay in an aerial clash and completely missed the ball.
Huddersfield had another escape soon after – Lossl pushing away Sanchez's fierce 20-yard drive after the Chile star brilliantly made space for himself.
United had to wait until the early stages of the second half for their next clear chance, as Sanchez reached Lukaku's knock-on and saw his effort blocked by Tommy Smith.
However, Huddersfield's resistance was finally brought to an end just before the hour.
Mata curled a fine cross in towards the near post from the left flank and Lukaku nipped in front of his marker to divert home.
Pogba was introduced from the bench for Lingard in the 65th minute and United's second goal soon followed.
Sanchez won a penalty after a clumsy challenge just inside the area by substitute Michael Hefele and, although his spot-kick was saved by Lossl, the rebound came straight back to the former Arsenal star and he swept it home with minimal fuss.
Proceedings petered out from there, as Mourinho's side comfortably held on to take all three points.
