The visitors got off to the perfect start in the second minute, when Steven Naismith’s drilled cross in to the six-yard box was tipped away by Julian Speroni into the path of Lukaku.
Palace responded immediately with Phil Jagielka clearing Dwight Gayle’s effort off the line, before the hosts lost Marouane Chamakh through injury after a clash of heads with Seamus Coleman.
The hosts continued to press in the second half in search of an equaliser, but rarely threatened Joel Robles in the Everton goal as Roberto Martinez’s side secured only their fifth clean sheet of the season.
Making just one change from the side that drew 0-0 at West Brom 12 days ago, Lukaku gave the visitors the early lead before the recalled Gayle came close to levelling moments later, beating Robles to the ball before lifting an effort towards goal, only to see Jagielka hack clear just in time.
The Eagles responded well to going behind, with Yaya Sanogo, making his home debut for the Eagles, having a low shot blocked and Chamakh having an effort deflected off John Stones.
Chamakh was taken off on a stretcher with a damaged nose midway through the first half, after a clash of heads with Toffees defender Seamus Coleman at the far post.
The home side continued to look livelier, with Jason Puncheon having a low, powerful shot tipped around the post from the edge of the area, while Jordon Mutch fired over in first-half stoppage time.
Palace started the second half brightly, with Sanogo narrowly avoiding getting on the end of Mutch’s powerful cross.
The hosts’ final ball was causing the Everton defence few problems, with Joe Ledley’s free kick easily claimed by Joel and Puncheon’s searching cross towards the far post headed clear by Coleman.
Wilfried Zaha was introduced on 65 minutes and instantly won two corners, while Glenn Murray was also brought on by Alan Pardew as the hosts searched for a way back in to the game.
With Palace pushing players forward, Everton had chances to double their advantage, with Speroni forced into a smart save to palm away Aiden McGeady’s curling strike towards the far corner.
Lukaku drilled an effort into the side-netting after Naismith’s pass into the area, while McGeady fired an effort wide from a tight angle in injury time.
Crystal Palace: Speroni (7), Ward (6), Delaney (5), Dann (6), Kelly (6), McArthur (7), Ledley (6), Puncheon (6), Chamakh (6), Sanogo (5), Gayle (7).
Subs Used: Mutch (7), Zaha (7), Murray (6).
Everton: Joel (7), Coleman (7), Stones (8), Jagielka (7), Baines (7), Besic (6), Barry (6), Mirallas (7), Naismith (7), McGeady (7), Lukaku (6)
Subs Used: Oviedo (6), Kone (6)
Man of the match: Phil Jagielka