Darren Moore’s main focus is to guide the Owls back to the Championship at the first time of asking after their nine-year stint in the second tier ended amid financial troubles last season.
But the former Doncaster Rovers boss presided over the club’s first first-round exit in the coveted cup competition since conceding four at home to Swindon Town in 2004.
The South Yorkshire side held Argyle to a goalless draw at Hillsborough nine days ago.
But the League One leaders secured a trip to Rochdale thanks to Swansea City loanee Jordan Garrick’s brace, either side of former Rangers’ younger Ryan Hardie’s effort.
The Owls made a bright start to proceedings as Florian Kamberi forced Mark Cooper into an early save but it was the hosts who drew first blood in the 20th minute against the run of play.
Moore’s men were caught on the counter as Garrick controlled Panutche Camará’s cross well before firing the home side ahead.
Argyle continued to break at pace and catch the visitors out. This time Joe Wildsmith did well to keep out Conor Grant after he was found by Garrick.
And it was two nine minutes before the break as Hardie doubled the advantage when his effort went in off the post.
Wildsmith kept the deficit at just the two as he once again kept out Grant as Wednesday struggled to make a mark in the final third. The second half lacked any real impetus until Plymouth pounced to put the game beyond doubt.
Any hopes of a second-half fightback were crushed with Garrick’s second with 23 minutes left. Wildsmith got himself into all sorts of trouble from a ball over the top at the 23-year-old calmly slotted home his third goal of the season.
The keeper kept his head as he produced a fine save to tip substitute Kieran Agard’s effort past the post late on.
Agard thought he had scored from the resultant corner after Wildsmith fumbled, before the referee blew for a foul on the keeper.
Plymouth Argyle: Cooper, Wilson, Scarr, Gillesphey, Edwards, Pereira Camará, Houghton (Randell, 84), Broom (Mayor, 76), Grant, Garrick (Agard, 71), Hardie. Unused subs: Cooper, Pursall, Law, Shirley, Jenkins Davies, Burton.
Sheffield Wednesday: Wildsmith, Paterson, Dunkley, Palmer, Adeniran (Sow, 28), Dele-Bashiru, Wing, Brown, Bannan (luongo, 74), Kamberi (Shodipo, 75), Berahino (Corbeanu, 59). Unused subs: Peacock-Farrell, Byers.
Referee: C Sarginson (Staffordshire)