West Brom maintained their push for a European spot but needed an own goal from Swansea midfielder Jonathan de Guzman to seal victory in the Barclays Premier League clash at the Hawthorns.
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Capital One Cup winners Swansea started in impressive style and deserved to go ahead through former West Brom striker Luke Moore's header.
Lukaku equalised with his 13th goal of the campaign before the decider from De Guzman but the visitors had a late effort from substitute Roland Lamah harshly ruled out for offside.
Lukaku could afford the luxury of having a penalty saved by Swans keeper Michel Vorm when the score was 1-1 as Albion secured their third win in the last four games.
The on-loan Chelsea player again impressed with his pace and power in the lone striker role.
His performances must surely be giving his parent club food for thought in deciding whether to recall him for next season or allow him to go back out on loan.
But even he was eclipsed by another excellent performance in the centre of the park by James Morrison, who pulled the strings to telling effect.
Swansea started in confident fashion and Michu missed an early opportunity to break the deadlock.
Pablo Hernandez's right-wing cross dropped invitingly for Swansea's 19-goal leading scorer inside the danger area but he sent his shot over the bar.
Albion right-back Billy Jones brought down Wayne Routledge 20 yards out but Jonathan de Guzman curled his free-kick past the post.
Michu squandered another golden opportunity after 19 minutes after a typical free-flowing move.
Good link-up play by Hernandez and Luke Moore carved the opening for Michu but he stabbed his shot wide with only Ben Foster to beat.
It came as no surprise when Moore celebrated his first start since Boxing Day by opening the scoring against his former club.
Moore got in front of his marker to glance his header from De Guzman's corner past Foster into the far corner of the net.
But Albion hit back and Lukaku levelled after 40 minutes with his 13th goal of the campaign.
Dorrans threaded a low ball towards the near post and Lukaku was first to react in steering the ball past Michel Vorm.
Jones was booked for bringing down Michu although Albion fans booed the Swansea player after believing he had dived.
Albion ended the first half on top and Vorm parried a fierce volley from Morrison.
Lukaku was prominent at the start of the second period and he shot wide after a mistake by Monk had allowed him a free run at goal.
The Belgium international then teed up a chance for Brunt, whose shot was saved by Vorm.
But he blotted his copybook after Albion were awarded a 56th-minute penalty for a challenge on Morrison by Routledge just inside the box.
Up stepped Lukaku to take the spot-kick but Vorm was equal to his low drive away to his right and was able to save.
The Baggies were not to be denied and an own goal from De Guzman put them ahead four minutes later.
Gareth McAuley's header from a Brunt corner was cleared off the line by Angel Rangel but rebounded back off De Guzman into the net.
Dorrans had a fierce drive deflected wide as Albion looked to extend their advantage.
Then Lukaku was just off target after creating space inside the Swansea box before being substituted to a standing ovation with nine minutes left.
Swansea substitute Roland Lamah had the ball in the net in the 87th minute.
But it was ruled out for offside although the ball appeared to hit home defender Gareth McAuley before rebounding into his path.