The hosts needed just four minutes to break the deadlock against the Toffees when Antonio Valencia's stunning strike ripped into the back of the net, but the majority of the match was a lot more even than the final scoreline suggests.
It wasn't until the final 10 minutes that United started to really pull away from their visitors, with Henrikh Mkhitaryan, Lukaku and Anthony Martial adding late goals to hand Jose Mourinho's side a flattering margin of victory.
The result sees United move back level with Manchester City on 13 points at the top of the table, whereas Everton have now lost each of their last four games without scoring across all competitions and will end the weekend in the Premier League's relegation zone.
All eyes were on Wayne Rooney in the buildup to the match as the Everton striker made his first return to Old Trafford since leaving in the summer following a trophy-laden 13-year stay which saw him become the club's all-time record goalscorer.
The spotlight was quickly stolen by Valencia, though, as the United full-back blasted his side ahead after just four minutes with an early contender for goal of the season, unleashing a thunderous first-time half-volley past Jordan Pickford from the edge of the area.
It was another dreadful start to the game for Everton, who found themselves three goals down before half time during their midweek Europa League defeat in Italy, but the Toffees did eventually settle into the game and almost levelled things up midway through the first half when Rooney steered a low first-time effort wide of the target having been found by Cuco Martina.
While Everton boasted United's all-time leading scorer in their ranks, the hosts also had Everton's highest ever Premier League scorer in the shape of Lukaku, and the Belgian should have continued his fine start to life at United when he spurned a glorious chance 26 minutes in.
Lukaku had another sight of goal towards the end of the first half, but a combination of Phil Jagielka and Pickford did enough to deny the £90m man as he continued to be frustrated.
Jagielka was needed again moments later to deny Eric Bailly at the near post from the resulting corner as Everton held on until half time despite growing United pressure towards the end of the opening 45 minutes.
It was Everton who began the second half in the ascendancy, though, and they should have been level within a minute of the restart when Rooney squandered another chance to mark his return with a goal as he fired a shot straight at David de Gea from close range having got the better of Bailly inside the area.
The visitors continued to hold their own in the opening stages of the second half, but United still had promising moments in the final third and both Marcus Rashford and Lukaku failed to make the most of space inside the Everton box.
Ronald Koeman's side had another clear chance to level things up shortly after the hour mark when Phil Jones's attempted clearance only steered the ball into the path of Gylfi Sigurdsson, who had two bites at the cherry but could not squeeze his effort past De Gea.
Having struggled to get going in the second half, United were just the width of the post away from doubling their lead with 25 minutes remaining when Mata curled a free kick around the wall and against the upright.
Everton were still in the game at 1-0, but United finally put the result beyond doubt with six minutes of normal time remaining when Mkhitaryan doubled their advantage following a mistake from Ashley Williams.
Williams's attempted pass out from defence was blocked by Marouane Fellaini and subsequently fell to Lukaku, who in turn played the ball through for an unmarked Mkhitaryan to sweep past Pickford from inside the area.
It was a goal which sparked a late flurry from United as Everton collapsed in the closing stages, with Lukaku getting his own name on the scoresheet five minutes later when he stroked the ball home from close range after a cross from the left had fallen at his feet.
Lukaku proceeded to celebrate in front of the Everton fans having become only the third player to scored five goals in his first five Premier League games for Everton, in addition to becoming the fourth-fastest player to reach 90 goals in the competition.
Everton almost got in on the act themselves during the action-packed closing stages, but De Gea was alert to tip Sandro Ramirez's powerful long-range strike over the crossbar before United added their fourth in the second minute of stoppage time.
Martial's magic footwork saw him escape two Everton defenders and break into the box, where Morgan Schneiderlin handled the ball while trying to stop his compatriot's mazy run. Martial took the spot kick himself and sent Pickford the wrong way to cap off the scoring and complete another impressive home win for his side.
The Red Devils have now won five consecutive home games without conceding for the first time since April 2006, including four from four across all competitions so far this season.
Everton, by contrast, have now gone four games without scoring for the first time in more than a decade and have lost four games on the bounce for the first time since January 2015.
It is also the first time in 13 years that the Toffees have gone 11 Premier League away games without a win, with Koeman's side having won just once on the road in the top flight this calendar year.