The League Two Sulphurites produced a brave attacking performance at Hillsborough, causing their higher-division hosts all kinds of problems for an hour of the contest.
But, having failed to find the back of the net despite managing 10 shots on target while the game was still in the balance, Simon Weaver's men were brutally taken apart as the Owls showed them how its done in front of goal.
The visitors began extremely brightly, Danilo Orsi going through on the Wednesday goal inside the first minute but only managing to roll a tame shot straight at Joe Wildsmith.
Moments later, Jack Diamond embarked on a run down the left, beat a couple of defenders and forced Wildsmith into a decent stop at his near post.
The Owls then went down the other end and Massimo Luongo thudded an effort from distance against the upright.
Town lost Jack Muldoon to injury soon afterwards, and suffered another blow on 17 minutes when the hosts broke the deadlock.
Ryan Fallowfield played a hospital pass to Josh Falkingham in midfield, gifting Wednesday possession and enabling them to attack down their left.
From there, Olamide Shodipo picked out Jack Hunt at the far post, and although he missed his kick initially, the Owls skipper had plenty of time to steady himself and tee-up the unmarked Saido Berahino for the simplest of close-range finishes.
Harrogate's response to falling behind was fairly positive, and they came within a whisker of levelling matters in the 24th minute.
Alex Pattison led a swift counter-attack with a driving run down the middle of the pitch, picking out Diamond, who saw an effort blocked. Orsi picked up the rebound and beat Wildsmith but the ball came back off the base of the post.
Lewis Page then marauded forward from left-back and found Aaron Martin in a great position, though the substitute failed to beat Wildsmith from close range.
Soon afterwards, Simon Power was felled inside the penalty area by a challenge from behind, though referee Trevor Kettle turned his back on appeals for what looked a clear spot-kick.
Visiting custodian Mark Oxley was then required to come off his line and save from Berahino with his legs, though it was Town who finished the half the stronger.
Orsi's clever touch got Martin in behind down the right, though his finish was once again poor and straight at Wildsmith, who had to make a much more difficult save from Pattison's half-volley just seconds later.
The second period began with Diamond setting off on a superb weaving run down the left which opened up the Owls' back-line. The Sunderland loanee went past three men as if they weren't there, but then blasted straight at Wildsmith.
Town were looking the more likely at this stage, however they fell further behind when Shodipo delivered another inviting centre from the left and George Byers glanced a 57th-minute header just inside the far post.
Undeterred, Harrogate again reacted positively, Burrell drawing an excellent reaction stop out of Wildsmith with a near-post header from Falkingham's right-wing cross before Pattison burst into the area and drilled a strike against the legs of the overworked home goalkeeper.
But, barely seconds later, another devastating counter-attack saw Sylla Sow put the game beyond the Sulphurites with a neat finish across Oxley and into the bottom corner from the left-hand edge of the box.
Things went from bad to worse on 67 minutes as centre-half Connor Hall was dismissed for a professional foul on Korede Adedoyin and the Wednesday substitute converted the resulting spot-kick.
With 10-man Town very much in survival mode by this stage, they managed to make it through to the final whistle without taking any further punishment, though Oxley was required to make an excellent one-on-one save to prevent Sow from making it 5-0 late on.
Tuesday's result means that the Owls top Group H, with Town qualifying for the knock-out stages as runners-up.
Sheffield Wednesday: Wildsmith; Hunt, Brennan, Agbontohoma, Brown; Luongo (Dele-Bashiru), Byers (Adeniran 73); Waldock; Berahino (Adeyoyin 61), Shodipo; Sow. Unused substitutes: Render, Paterson, Corbeanu, Wing. Unused substitutes:
Harrogate Town: Oxley, Fallowfield, Burrell, Hall, Page; Power, Falkingham, Pattison, Diamond (Thomson 72); Orsi (Smith 72), Muldoon (Martin 8). Unused substitutes: Cracknell, Wilson, Williams.
Referee: T Kettle (Rutland).