After ex-Cumbrians loanee Ben Marshall gave Owls a first-half lead, goals from Lee Miller, Rory Loy and Liam Noble in a 15-minute spell after half-time left the visitors - who finished with 10 men as Gary Madine was dismissed for a second yellow - reeling.
Marshall's 35th-minute strike for Wednesday was a lesson in how to create space and apply a clinical final blow. Cutting in from the right, he beat two defenders before rifling a low left-foot shot wide of the diving Adam Collin from 20 yards.
Carlisle raised the tempo after the break and Miller's sixth goal in 10 games put them on terms in the 50th minute. Loy headed James Berrett's cross back across goal and Stephen Bywater's touch deflected the ball into the path of Miller, who slid it home from close range. Six minutes later, it was 2-1 to Carlisle as Loy blasted home from 16 yards after Miller had set up Noble's pass.
Wednesday were further behind in the 65th minute when Noble made it three, coming in from the left to coolly pick his spot and beat Bywater after Miller and Tom Taiwo had done the spadework. It was an unhappy return for Madine when the ex-Carlisle striker was sent off in the 81st minute after a challenge on James Tavernier.
Skipper Rob Jones headed a second for Owls in the 89th minute from Marshall's free-kick, but it mattered little.