The home side took the lead at a snowy Brunton Park on the half-hour when Francois Zoko supplied the vital near-post header from Gary Madine's cross with the ball bouncing in off the foot of the left-hand post.
The Owls lost Neil Mellor, their hat-trick hero against Hartlepool in the previous round, two minutes before the break after an innocuous looking challenge with Lubo Michalik.
Carlisle, last year's beaten finalists, went further ahead on the hour when Michalik thundered home a volley from 12 yards after Wednesday failed to clear James Berrett's corner.
Ben Marshall capped a blistering spell of attacking in the 69th minute, delicately flicking the ball over the advancing Nicky Weaver from Madine's cross.
Darren Purse pulled one back for the visitors in the 84th minute with a near-post header from Tommy Miller's corner but their six-game winning run came to an end.