Tom Davies poked in for the hosts after 30 seconds, before Leicester rallied with a slick counter-attacking goal from Islam Slimani and a superb free-kick from Marc Albrighton.
But Leicester, much changed before Wednesday’s Champions League quarter-final at Atletico Madrid, never looked comfortable.
Lukaku headed in a Ross Barkley cross to draw Everton level and Phil Jagielka was allowed too much space at a corner as the Toffees reached the break ahead.
Lukaku drilled in from close range after the interval and the withdrawal of the bright Demarai Gray and striker Jamie Vardy seemed to signal Leicester’s attentions turning towards Europe.
With his 22nd and 23rd Premier League goals putting him four clear of Harry Kane in the top-flight scoring charts, Lukaku will soon have to search further afield for his scoring benchmarks.
His game-sealing second – a sharp finish after cleverly lurking in the shadow of Jagielka at a corner – means that only five other men have scored more in Europe’s top four leagues.
The last time that Everton had such a prolific goalscorer was Gary Lineker in the 1985-86 season. Lineker left at the end of that campaign for Barcelona and, after Lukaku turned down a £140,000-a-week contract extension last month, it seems European football is the minimum required to keep him at the club.
Victory moves Everton level with sixth-placed Arsenal, below them on goal difference, albeit having played three games more.
Seventh should be good enough for a Europa League place, however, with Manchester United winning the League Cup and the FA Cup semi-finals being contested by teams above them in the table.
Madrid on the mind for Foxes
Victory for Leicester at Goodison Park would have extended their winning streak in the league to six matches – more than they achieved at any point in last season’s title-winning campaign.
However, their title defence, fatally undermined by their form under Shakespeare’s predecessor Claudio Ranieri, has now been eclipsed by their run to the last eight of the Champions League.
The return of the rested Wilfred Ndidi in midfield should add much-needed steel to the side for a testing evening in the Vicente Calderon as they aim for silverware that would arguably surpass even last season’s Premier League coup.