Kelechi who is now Nigeria's record goal scorer in the English Football Association Challenge Cup (FA Cup), had not yet found the back of the net in the Premier League since he moved from Manchester City.
Ben Foster nearly conceded in comedic fashion when he fired a clearance against centre-back Ahmed Hegazi that rebounded back towards his own net.
There appeared to be minimal danger when Mahrez floated forward a sumptuous pass from just inside his own half. Craig Dawson allowed Mahrez's lofted through ball to drop over his head and Vardy finished superbly with a left-footed volley.
It sucked the life out of the hosts and they escaped after 36 minutes when Vardy poked in from eight yards only to be correctly ruled offside.
The Swans had Jordan Ayew sent off in the 11th minute.
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Substitute Iheanacho picked out Mahrez, who was not tracked, and he coolly lobbed Foster.
Iheanacho's 76th-minute header made it 3-1, prompting numerous home fans to head towards the exits.
Iborra rubbed it in by heading in a Marc Albrighton corner in the stoppage time.
The result means that West Brom are now 8 points from safety and will require nothing short of a miracle to survive this season.
There was a buzz about Albion and they were comfortable until the Foxes levelled after 21 minutes with a goal of brilliance and simplicity.
The home side would've needed to make history if they were to improve their survival hopes in the game, as they had never won in the seven Premier League home meetings between these sides - with the Foxes picking up five wins and two draws.