Lukaku scored a hat-trick for the Toffees after Callum Patterson had given Hearts an early lead.
Hearts made a bright start to the match and took the lead after just nine minutes when a corner from Kevin McHattie was powered into the net at the back post by Patterson.
And the home side threatened again three minutes later when Sam Nicholson timed his arrival perfectly to meet a Morgaro Gomis cross, but the midfielder sent his glancing header wide.
Everton hit back on 18 minutes after Neil Alexander kept out a close-range header from Lukaku with a brilliant save before Alim Ozturk was adjudged to have used his hand to block the rebound from Gareth Barry, and the big striker stepped up to dispatch the penalty.
The visitors picked up the pace and Leon Osman had a shot deflected over the bar after he weaved into the Hearts penalty area moments before he headed wide from the resulting corner.
But Lukaku scored his second goal 10 minutes before the break when he fired home from close range following good work from Seamus Coleman.
Moments later Lukaku crashed a shot inches over the bar and then he took advantage of slack defending to release Kevin Mirallas, but he dinked the ball inches wide of goal.
Everton did add a third four minutes after the interval from another Lukaku spot-kick, and this time the award was more clear cut as McHattie upended Mirallas.
Hearts could not cope with Lukaku and he fired another effort wide when Arouna Kone dragged a low ball across the box.
Three minutes later, Lukaku was denied again when his low shot from inside the penalty area forced Alexander into a good save.
Everton continued to press and Mirallas had a shot cleared off the line by Paterson shortly before Osman sent a header wide.
Alexander produced another impressive save to deny Lukaku and Everton a fourth goal following good play from Kone.
Hearts had the last real chance of the match four minutes from time when Billy King created space with some neat footwork to test Time Howard from the edge of the box, but the rebound was smashed over the bar by Liam Smith.