Koren, a half-time replacement for Jerome Thomas, stuck with a 30-yard drive in the 79th minute, as Albion finally found a way past the inspired Owls goalkeeper Lee Grant.
He had pulled off four stunning second-half saves and his heroics looked set to earn Wednesday a draw before Koren struck in spectacular fashion to move Albion two points clear of Nottingham Forest.
The Slovenian's goal earned Albion their first victory over Wednesday at The Hawthorns since 1975 as they answered the demands from manager Roberto Di Matteo who had labelled the match a 'must-win game'.
Wednesday will feel hard done by as they looked set to hold out for a point but now remain rooted in the relegation zone.
It needed something special to beat Grant as he had pulled off three top-class saves in the space of five minutes - tipping away a Jonas Olsson header on the hour, denying Ishmael Miller five minutes later and from the resulting corner he then saved another headed effort - this time from James Morrison.
The full value of Grant's saves should then have been exploited by Leon Clarke only for the former Wolves striker to be denied at the near post by goalkeeper Scott Carson in the 69th minute.
That was a pivotal moment for Wednesday, but when Grant then saved from Graham Dorrans after 74 minutes they still looked odds on to collect a share of the spoils.
However, all of Grant's good work ultimately counted for nothing when Koren tried his luck from long range.
The frantic spell of action was a far cry from a low key first half when neither goalkeeper was forced into a save.
Albion did carry the greater threat with Wednesday old-boy Chris Brunt volleying wide and Morrison and Dorrans both failing to hit the target when well placed.
But Wednesday's pace worried Albion on the break and they were unfortunate not to cash in on that in the 37th minute.
Jermaine Johnson used his speed and power to muscle his way past Marek Cech and was then faced with only Carson to beat.
Instead of bearing down on goal, Johnson tried to beat Carson with a chip and while he did that, the ball also flew over the bar.