Stamford Bridge, the home of Chelsea, will be the Owls’ destination a week on Sunday for an FA Cup fourth-round tie that will be broadcast live to the nation on BBC1.
Atdhe Nuhiu’s strike early in the second half was enough to settle a keenly-fought replay at Luton Town in the favour of the Championship club.
Over the course of 180 minutes a side led by caretaker manager Steve Agnew just about deserved to go through.
But the evening was not without its nervy moments for the Owls and their 1,473 travelling supporters as Dan Potts struck a post and Cameron Dawson made a couple of vital saves during a second-half onslaught from the League One promotion hopefuls.
Come the final whistle, however, it was the visitors celebrating the booking of a trip down the King’s Road at the end of an evening that had begun with a potentially fatal Brexit defeat for the Prime Minister 35 miles to the south in the House of Commons.
May will face a vote of no confidence today, meaning in theory she could soon be calling on the Queen to ask for Parliament to be dissolved in readiness for a General Election.
Wednesday’s own adventure in the capital on Sunday week will be a wholly more enjoyable affair, as the south Yorkshire club make a first visit to the Bridge since the turn of the Millennium.
Both sides had started the night displaying all the conviction of the Government when losing by a colossal 230 votes in Parliament just a couple of hours earlier.
Mistakes abounded as move after move broke down just when things looked to be getting interesting.
As if to underline how disjointed proceedings became during that first half the closest Luton came to a breakthrough was when Tom Lees’s back pass caught goalkeeper Cameron Dawson unawares so badly that the ball fizzed past him en route to bouncing a couple of yards wide.
There was also a clumsy tackle from Nuhiu early on that handed the hosts a free-kick right on the edge of the penalty area that Elliot Lee could only fire into the Owls’ defensive wall.
Wednesday’s best moment of the opening 45 minutes was slightly more impressive, Barry Bannan’s left-footed piledriver being deflected against a post by Steven Fletcher’s head.
But even that came almost by accident, Bannan’s effort being so fierce that his fellow Scot was not the intended target, but James Shea’s goal.
The tie needed a lift and it came so early in the second half that some in the 9,259 crowd had yet to retake their seats.
Just 69 seconds had elapsed when the visitors went ahead. A sweeping move by the visitors caught out the Hatters’ backline, allowing Liam Palmer to gallop clear down the right flank.
His cross found Fletcher, whose first-time shot struck Nuhiu in the mid-riff. Quick as a flash, however, the big Kosovan tried his own luck and this time the ball flew beyond Shea via a Hatters leg.
Buoyed to be ahead, the Owls pushed for a second goal as George Boyd fired just wide from 25 yards and then Adam Reach did the same.
Reach also had a shot well saved by Shea low to his right after a swift counter attack involving a multitude of Wednesday players.
Luton, to their credit, made sure the Yorkshire side could never rest with James Justin bringing a fine save from Dawson on the hour.
Nine minutes later the Wednesday goalkeeper had to react smartly to deny Dan Potts after the substitute had skipped past Palmer before unleashing a shot that was turned behind for a corner.
Potts then went even closer with a downward header that bounced across the six-yard box before striking a post.
Justin also had a big appeal for a penalty turned down by referee Tim Robinson after theatrically going down amid a succession of Owls players hacking at the ball in a desperate attempt to clear their lines.
It proved to be Luton’s last stand as Wednesday booked that glamorous trip down the King’s Road.
Luton Town: Shea; Stacey, Pearson, Bradley, Justin; Mpanzu, Berry (Potts 68), Shinnie, Lee (Jarvis 88); Collins, LuaLua. Unused substitutes: McCormack, Jones, Isted, Sheehan, Neufville.
Sheffield Wednesday: Dawson; Lees, Hector, Thorniley; Palmer, Bannan, Reach, Boyd, Fox; Fletcher (Matias 89), Nuhiu. Unused substitutes: Jones, van Aken, Winnall, Baker, Wildsmith, Pudil.
Referee: T Robinson (West Sussex).