SUPER-SUB Robert Koren blasted Albion back into the top two with a late thunderbolt.
The Slovenia skipper, a half-time replacement for Jerome Thomas, broke the deadlock in the 79th minute with a wonderful left-footed strike to seal the Baggies' first home win over the Owls for 35 years.
Albion's fourth successive Hawthorns league victory was enough to leapfrog Nottingham Forest into second place, moving two points clear of Billy Davies' boys.
Roberto Di Matteo's men weren't at their best but would have got on the scoresheet much earlier had it not been for a virtuoso second-half display by Wednesday keeper Lee Grant.
Luke Moore and Graham Dorrans earned recalls as Di Matteo rang two changes, with Simon Cox and Ben Watson dropping to the bench.
New signing Giles Barnes was also named among the subs for the first time after his long Achilles injury lay-off, with Joe Mattock making way.
Wednesday made one change as Luke Varney replaced Marcus Tudgay while two ex-Baggies - skipper Darren Purse and midfielder James O'Connor - were in the Owls' starting XI.
Albion showed their support for injured team-mate Gonzalo Jara's homeland, which is recovering from a massive earthquake, by wearing 'Help Chile' logos on their shirts.
Both teams appeared to be struggling for confidence as the game got off to a tentative start.
And it wasn't until the 13th minute that there was even a hint of goalmouth action, when Tommy Spurr had to make a frantic far post clearance to prevent Dorrans' corner dropping over the line.
Chances remained at a premium until the 20th minute when former Wednesdayite Chris Brunt smashed an acrobatic volley just wide after James Morrison had helped on a Marek Cech centre.
The visitors were still struggling to get anywhere near the Baggies goal when Moore towered on the edge of the Wednesday box to direct a long Gabriel Tamas punt into Morrison's path.
Unfortunately, the Scotland international could only volley weakly wide from 18 yards.
The Owls recorded their first moment of danger on 34 when the lively Jermaine Johnson whipped in a low cross from the left and Tamas had to slide in at the last minute to prevent Varney from threatening Scott Carson's goal.
Albion immediately broke up-field but Dorrans was well off target with a 20-yard drive following a jinking Cech run.
However, against the run of play, Wednesday carved out the game's first gilt-edged chance in the 37th minute when Johnson shrugged off Cech to latch onto Varney's ball over the top and race clear.
The Wednesday midfielder only had Carson to beat but saw his audacious chip loop onto the roof of the net.
Jonas Olsson almost gifted Tommy Miller a clear run at goal when he inadvertently trod on the ball and lost his footing.
But the vice-skipper recovered brilliantly to slide in and dispossess the former Sunderland man before he could shoot.
The final action of a poor half saw Moore volley wide after the former Villa striker had done well to outmuscle Purse to Tamas' long aerial ball.
Di Matteo made a change at half-time, with Koren replacing Jerome Thomas.
Neither team managed a shot on target during the first 45 and that trend continued when the second half was only 20 seconds old, with Leon Clarke miscuing wide from O'Connor's lay-off.
Albion were failing to capitalise on their lengthy spells of possession and Johnson was the next to try his luck, firing wide from 25 yards.
The Baggies finally recorded another effort at goal on 56 when Moore saw his quick-fire turn and shot in the box blocked by Purse after neat play by Morrison.
Di Matteo replaced Moore with Ishmael Miller two minutes later and the sub was soon in the thick of it, shrugging off O'Connor before dragging his shot wide from 25 yards.
Albion finally got some momentum going and Owls keeper Grant had to pull off a stunning point-blank block to keep out Olsson's downward header from Dorrans' pinpoint centre.
Miller eventually got the rebound under control, only to hack over from 15 yards.
Grant had to be at his best again on 65 to turn wide Miller's downward header from Brunt's far post cross.
From Dorrans' ensuing corner, Grant pulled off another brilliant stop, tipping over Morrison's bullet header from point-blank range.
A goal just would not come and it almost got worse for Albion when Carson brilliantly denied Clarke in a one-on-one after the former Wolves striker had skipped past home debutant Steven Reid.
Back came Albion and Morrison saw a 20-yard drive fly off Eddie Nolan's outstretched leg and loop just over.
Grant's one-man show continued on 74 when he did well to tip over Dorrans' 30-yard thunderbolt.
Poor Albion defending presented a half-chance for Clarke moments later, but the striker fired well over from 18 yards.
The hosts had been much improved after the break but another opportunity went begging when Miller headed weakly wide from Olsson's delightful ball over the top.
But a goal had been coming - and boy did it arrive 11 minutes from time.
Koren cut in from the right and smashed an unstoppable left-footed shot into the far left-hand corner from 25 yards.
It was a goal worthy of winning any game and so it proved, with Albion seeing out the remainder of the match with little trouble.
ALBION (4-5-1): Carson; S Reid, Tamas, Olsson, Cech; Brunt, Morrison (Barnes 83), Mulumbu, Dorrans, Thomas (Koren HT); Moore (I Miller 58). Subs not used: Kiely (gk), Meite, Watson, Cox.
SHEFFIELD WED (4-4-2): Grant; Spurr, Purse, Beevers, Nolan; Johnson (Soares 72), Potter, O'Connor (Jeffers 83), T Miller; Varney, Clarke. Subs not used: O'Donnell (gk), Hinds, McAllister, Simek, Gray.
GOALS: ALBION - Koren (79).
BOOKINGS: WEDNESDAY - Spurr (foul 31).
REFEREE: S Mathieson (Cheshire).
ALBION STARMAN: Gabriel Tamas.