Saido Berahino and Craig Dawson had put the Baggies 2-0 ahead but Lukaku hit back immediately after Dawson's header nine minutes after the break.
The 22-year-old scored 17 goals during an impressive loan spell at Albion from Chelsea in 2012-13 and ignited Everton's challenge with Kone's impact also crucial, the forward scoring within three minutes of coming on as a second-half substitute.
It was the first time Everton had come back to win a game in the Barclays Premier League since May 1994 - when they beat Wimbledon 3-2 to stay in the top flight.
There was little hint of the drama to come, which left Albion winless at home in the Premier League this season, during a wretched and instantly forgettable first half which was only lit up by Berahino's opener.
Without the sidelined John Stones, out with a knee injury, Ramiro Funes Mori stepped in for his full league debut but the Everton defender was at fault for Albion's first goal after a dull first 25 minutes.
Only the visitors created a clear cut chance before the hosts struck when Ross Barkley blew his lines from eight yards after 29 minutes.
The England midfielder started the move with a neat dummy before the ball was worked to Gerard Deulofeu on the right.
His fine low pass dissected the Albion defence only for an on-rushing Barkley to wretchedly miscue his shot wide.
It was a horrible miss and one which the Toffees would pay for when the Baggies opened the scoring four minutes before the break.
Mori had enjoyed a untroubled half but imploded when he lost the ball near the halfway line after trying to run out of defence.
It found Morrison and his excellent throughball for Berahino saw the striker slip in a low shot via Tim Howard's leg.
It was the striker's second goal since returning to the starting line-up following his deadline day strike threat and underlined why the Baggies were so determined to keep him away from Tottenham.
Buoyed, James McClean fired over immediately after the restart and Mori escaped a penalty shout after pushing over Salomon Rondon.
Darren Fletcher then had a header turned needlessly wide by Howard after 54 minutes and Albion doubled their lead from the resulting corner.
Phil Jagielka and Barkley got themselves in a mess trying to pick up Dawson and Jonny Evans which allowed Dawson to nod in Chris Brunt's deep delivery.
Instantly, though, Everton hit back when Lukaku headed Deulofeu's cross past Boaz Myhill.
With that the game turned even if Everton's desperation was growing and James McCarthy was booked for an awful dive in the box with 20 minutes left.
Yet Kone levelled four minutes later - and just three minutes after coming on.
Again, Lukaku did brilliantly on the edge of the box to spot Kone's run and threaded the ball through to the former Wigan forward, who was just onside, to drill past Myhill.
The Baggies were shellshocked and former Chelsea striker Lukaku was not finished as he wrapped up Everton's comeback with six minutes left.
There was a hint of handball when he met Deulofeu's cross by splitting Dawson and James Chester but with Myhill slow to react Lukaku smashed in from close range to win it.
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