The Red Devils will officially have a new head coach on Monday, with Amorim arriving from Sporting Lisbon, and he will take over a side that have gone unbeaten in their four matches since Erik ten Hag's dismissal at the end of last month.
Bruno Fernandes was presented with a special frame ahead of kickoff at Old Trafford to mark his 250th Man United appearance, and he was on the scoresheet in the 17th minute, brilliantly picking out the bottom corner after being found by Amad Diallo, who entered the contest fresh from his brace against PAOK in the Europa League.
The Red Devils captain then helped to double his side's advantage in the 38th minute, nudging a cross from Noussair Mazraoui into the back of the net, albeit with the help of a deflection off Victor Kristiansen, and it went down as an own goal for the Leicester defender, who did not know too much about it.
Alejandro Garnacho then made sure of the points in the 82nd minute, with the forward brilliantly picking out the top corner from the edge of box to register his seventh goal of the campaign.
Van Nistelrooy has signed off with a record of three wins and one draw from four matches, but it remains to be seen whether there is a role for him at Old Trafford under Amorim's management, as the 39-year-old has already confirmed that he will be bringing his own staff with him from Sporting.
The result has moved Man United into 13th position in the Premier League table, just three points behind fifth-placed Chelsea, while Leicester remain 15th, three points above the bottom three, with the Foxes seemingly facing a relegation battle this term.
MAN UNITED VS. LEICESTER HIGHLIGHTS
17th min: Man United 1-0 Leicester (Bruno Fernandes)
Man United make the breakthrough in the 17th minute of the match, and it is the captain Fernandes who drills one into the bottom corner after being found by Amad, who expertly flicked the ball towards his teammate.
38th min: Man United 2-0 Leicester (Victor Kristiansen own goal)
Man United double their advantage in the 38th minute, as Fernandes nudges a cross from Mazraoui into the back of the net, but he had some help from a deflection off Kristiansen, and it goes down as an own goal.
Long way back for Leicester from his position. Are Man United heading for a huge Premier League win?
82nd min: Man United 3-0 Leicester (Alejandro Garnacho)
Man United secure all the points in the 82nd minute of the match, as Garnacho fires one into the top corner from the edge of the box - it is a wonderful finish from the Argentina international.
MAN OF THE MATCH - BRUNO FERNANDES
Fernandes has struggled to show his best form for much of this season, but the 30-year-old has now scored twice in his last two Premier League appearances, while he had a huge part to play in Man United's second against Leicester before setting up Garnacho to make it 3-0 in the latter stages.
The Portugal international also made four tackles during an all-action performance, while he provided six key passes, with Leicester struggling to deal with the attacker on his 250th appearance for Man United.
MAN UNITED VS. LEICESTER MATCH STATS
Possession: Man United 51%-49% Leicester
Shots: Man United 10-6 Leicester
Shots on target: Man United 2-5 Leicester
Corners: Man United 1-2 Leicester
Fouls: Man United 8-3 Leicester
BEST STATS
WHAT NEXT?
Life under Amorim will begin for Man United against Ipswich Town on November 24, with the Portuguese set to have his first taste of English football at Portman Road, while his first game at Old Trafford will be against Bodo/Glimt in the Europa League on November 28.
Leicester, meanwhile, will be aiming to return to winning ways when they welcome Chelsea to the King Power Stadium on November 23, before ending the month with a trip to Brentford.