Fulham netted a consolation effort from the spot through Aboubakar Kamara and had Andre-Frank Zambo Anguissa dismissed for two yellow cards as Claudio Ranieri's men were well beaten in the North-West of England.
Both teams started brightly at Old Trafford without posing much of a threat in either final third, but United were able to take the lead with the first shot of the match.
Young found it too easy to cut inside onto his right foot, and it gave the full-back the chance to drill an effort into the far corner from 15 yards out.
The goal acted as the catalyst for United to start showing more ambition in the final third, but after Chris Smalling had a header deflected narrowly wide of the target, Fulham made an impact at the other end with Aleksandar Mitrovic firing a low effort straight at David de Gea from distance.
The visitors remained in the match as long as it stayed at 1-0, but United extended their advantage before the half-hour mark when Rashford pulled the ball back for Mata to fire a low shot into the bottom corner from 14 yards out.
United were starting to play with a swagger but their third goal of the first half came via a well-worked set piece, which was finished off by Lukaku from close range after the Belgian had got on the end of a pass from Mata.
The home side went back on the offensive after the restart with Rashford seeing a well-struck effort saved by Sergio Rico, but Fulham began to offer something in the final third with Andre Schurrle curling just over from 20 yards out.
Ander Herrera should have put United further ahead on the hour mark but the Spaniard dragged a 10-yard effort wide of the far post, and that miss gave the Cottagers a chance to get back into the match.
Herrera was adjudged to have fouled Kamara inside the area - despite winning the ball - and the Fulham substitute got to his feet to slot the spot kick into the bottom corner with De Gea diving the opposite direction.
However, Fulham were reduced to 10 men a minute later when Zambo Anguissa picked up a second yellow card for a foul on Rashford, although the midfielder made little contact with the United man.
That dismissal effectively ended Fulham's hopes of a comeback and it soon became a case of whether United would further extend their lead during the closing quarter of the contest.
Rico made smart saves to keep out both Rashford and Lukaku, but the goalkeeper was at fault for United's fourth with Rashford's effort from 20 yards being parried into the corner.
United continued to push for a fifth with Scott McTominay squandering an opportunity soon after coming on as a replacement, but Mourinho will be more than satisfied having witnessed his side dominate from start to finish at a time when three points were desperately required.
MANCHESTER UNITED (4-2-3-1): De Gea; Dalot, Jones, Smalling (Rojo 58'), Young; Herrera, Matic; Mata, Lingard (Fred 73'), Rashford (McTominay 84'); Lukaku
FULHAM (4-3-3): Rico; Odoi, Mawson, Ream, Bryan; Seri, Anguissa, Cairney (Vietto 46'); Schurrle (Cisse 73'), Mitrovic (Kamara 46'), Sessegnon