It was a sixth straight victory over all competitions for the 10th-placed Foxes (10-6-14), who dropped five consecutive Premier League matches before winning five in a row on the domestic front.
It looked as if Craig Shakespeare's men were in for a frustrating night as Sunderland (5-5-20) had chances to level things while down 1-0, as their best chance came through top scorer Jermain Defoe, but he could only strike the post for the visitors.
Sunderland's scoreless stretch has now reached six straight matches dating to a 4-0 win at Crystal Palace on Feb. 4. The loss added to the current misery surrounding the club after manager David Moyes apologized for an inappropriate remark to a female reporter during an interview following a match last month.
Earlier Tuesday, the Sunderland brass said Moyes would remain manager even though the club is headed for relegation.
Marc Albrighton assisted on both goals for Leicester, which surrendered 15 shots to Sunderland but only three were on target.