Playing his first game in three weeks, Canice saw his side fall behind in the 25th-minute, goalkeeper Louis Gray with the unfortunate own-goal. The Cumbrians pulled themselves level on the stroke of half-time through Doug Loft.
With a draw meaning exit from the competition in the group stages, Canice was the Carlisle hero three minutes from time, crashing home a 30-yard rocket to earn Steven Pressley's side all three points and keep qualification hopes in their own hands.
The Cumbrians return to League Two action on Saturday, facing a monster trip to Plymouth Argyle.