Lukaku's 81st-minute goal proved the difference as West Ham's mini revival came to end following a run of four wins and one draw in their previous five Premier League games prior to the trip to Merseyside.
The host controlled the majority of the first half and went close through Steven Naismith and Gerard Deulofeu, but neither could find a way past Adrian in the West Ham goal.
Steven Pienaar hit the post straight after the restart and Sam Allardyce's men looked set to clinch a point and their 14th clean sheet of the Premier League season before Lukaku struck nine minutes from time to send Everton back above Manchester United in the table.
Much of the build-up was focused on whether the two sides' respective first-choice forwards would start, but Lukaku was only deemed fit enough for a place on the bench, while Andy Carroll sat out the first half-hour after his three-match ban.
Although Leon Osman nearly turned in Naismith's headed front-post flick-on, clear chances were scarce during the first 20 minutes, despite Everton generally dominating proceedings.
Naismith forced Adrian into his first save 22 minutes in as he fired at goal from an acute angle, but the Spaniard make a fairly straightforward stop.
With West Ham offering virtually no threat in attack, Allardyce made the decision to introduce Carroll for Carlton Cole after just 28 minutes, but he made little difference initially.
Deulofeu was the next to go close for Everton nine minutes before the break.
The on-loan Barcelona winger twisted and turned his way past a couple of defenders on the edge of the area before curling a left-footed effort towards the bottom-left corner, but the attempt went agonisingly wide.
Everton made a ferocious start to the second half and within two minutes of the restart Pienaar was presented with a massive chance as he latched on to Leighton Baines' throughball, but his first-time effort came back off the right-hand post.
John Stones enjoyed a solid first half, but just before the hour he almost contributed to Everton's downfall as he sliced a clearance only as far as Carroll, but luckily for the England Under-21 international, Tim Howard palmed the subsequent half-volley over the crossbar.
West Ham were doing a good job of keeping Everton at bay, but with nine minutes to go Baines cut a pass back to Lukaku and the Belgian hit an emphatic left-footed finish past Adrian to move the hosts up to sixth.
