New signing Mark Robins (pictured) scored his first brace for The Owls after Graeme Lee had given Chris Turner's side a 1-0 lead at the break, as Wednesday denied The Cumbrians another chance of reaching the final in Cardiff.
Scunthorpe United reached the next stage following a 1-0 silver goal victory at Third Division rivals Bury.
After a goalless 90 minutes at Gigg Lane, Paul Hayes's strike nine minutes into the first period of extra time was enough for Brian Laws's men to go through.
Matthew Blinkhorn's strike in the 109th minute of extra time was the difference as Blackpool defeated Stockport County 1-0 at Edgeley Park.
The other Northern Section quarter final between the only Conference side left in the competition - Halifax Town - and Division Three side Lincoln City was postponed at The Shay Stadium due to fog.
In the Southern Section quarter finals, struggling Southend United continued their good cup form with a 3-0 victory over Second Division side Luton Town.
Leon Constantine, Michael Kightly and Drewe Broughton scored the goals for The Shrimpers at Roots Hall.
Northampton Town needed a silver goal to defeat Barry Fry's Peterborough United 2-1 at The Sixfields Stadium.
After a 0-0 first half, Josh Low gave The Cobblers a 1-0 lead seven minutes after the break before Leon McKenzie equalised in the dying stages for The Posh, as both sides went into extra time.
In the first period of extra time, Lawrie Dudfield scored the Silver Goal winner ten minutes in to take Colin Calderwood's side through.
Finally, Colchester United needed the last minute of extra time to defeat Division Two rivals Wycombe Wanderers 3-2 at The Causeway Stadium.
The U's took a 2-1 lead into the break with a brace from Scott McGleish while Andy Thomson replied for Wycombe.
Tony Adams's side drew level after the break through Roger Johnson, as the match headed into extra time at 2-2.
In the final minute of that extra time, Jermaine Brown popped up to give Colchester a win and a place in the Southern Section semi-finals.
Click below for match summaries:
LDV Vans Trophy quarter finals
Northern Section
Bury 0-1 Scunthorpe (silver goal)
Carlisle 0-3 Sheff Wed
Halifax P-P Lincoln City
Stockport 0-1 Blackpool (AET)
Southern Section
Northampton 2-1 Peterborough (silver goal)
Southend 3-0 Luton
Wycombe 2-3 Colchester (AET)
