Ole Gunnar Solskjaer was without a host of first-team regulars due to injuries, but Lukaku stepped up having been given a rare opportunity to lead the line, his pair of goals playing a part in downing a spirited Palace side.
United just about shaded an even first half and went into the break ahead thanks to a fine Luke Shaw-inspired Lukaku goal in the 33rd minute.
Lukaku got his second of the day just after the break and, although Joel Ward pulled one back in the 66th minute, Ashley Young struck late on to end the hosts' hopes and keep United just a point behind fourth-placed Arsenal, who thrashed Bournemouth 5-1.
United should have taken the lead after just eight minutes, but Lukaku inexplicably volleyed over from eight yards after Shaw's pinpoint corner found the Belgium international unmarked.
Palace nearly capitalised a few moments later – Andros Townsend shooting over from inside the six-yard box after James McArthur's pass.
Lukaku atoned for his earlier miss just past the half-hour mark, producing a lovely right-footed finish into the bottom-right corner from 18 yards after a brilliant solo run from Shaw.
He doubled his tally seven minutes into the second half, knocking in from close range after Chris Smalling's header found Victor Lindelof, who in turn nodded the ball into the danger zone.
Palace got themselves a lifeline just after the hour – Ward scoring a diving header from Jeffrey Schlupp's delivery after Ashley Young had been robbed of possession in the corner.
The home side piled men forwards towards the end, but that ultimately brought their undoing as United caught them on the break seven minutes from time – Young finishing from a tight angle after being released by Paul Pogba.
What does it mean? Solskjaer and United pass a real test
Given United's injury problems and Palace's six-match unbeaten run coming into the game, this was never going to be a stroll. But United stood up to the task well, producing a workmanlike performance that was ultimately enough to remain well in the top-four hunt.
Palace, meanwhile, are now only five points clear of the bottom three.
Lukaku takes his chance
Ever since Solskjaer's arrival, opportunities to start as United's main striker have been few and far between for Lukaku, but he was excellent at Selhurst Park, getting involved in the general play and scoring two completely different goals.
Sanchez fails to sparkle
Sanchez was afforded a start due to United's injury crisis, but he offered very little, mustering just a single shot and failing to create any opportunities. He was unsurprisingly hooked for Marcus Rashford towards the end.
KEY OPTA FACTS:
- Palace have played more Premier League games without ever winning against United (D4 L16) than any side has against another in the competition's history.
- United are unbeaten in their last 22 league games against Crystal Palace, since a 3-0 away defeat in May 1991 (W18 D4).
- Roy Hodgson has lost more Premier League matches against United (12) than he has against any other side.
- Eight of Lukaku's 10 Premier League goals this season have been scored away from home - only Harry Kane (10) has netted more on the road in 2018-19 so far.
- Joel Ward ended a run of 86 Premier League appearances without a goal, since netting versus Chelsea in August 2015.