However, Sean O'Hanlon's own goal and a late strike from substitute El-Hadji Ba killed off any hope of an upset for the 5,631 travelling fans among a crowd of 21,973 and secured a fourth round home tie against either Kidderminster or Peterborough for Poyet's men.
The fact that it was achieved without skipper John O'Shea, Italy midfielder Emanuele Giaccherini and main striker Steven Fletcher will have been a source of comfort for the Uruguayan amid a cluttered schedule.
However, while progression in both domestic cup competitions has been welcome, it is the fight for Premier League survival which remains the overriding priority.
Carlisle have their own problems with the club sitting just three points clear of the drop zone, but they should return to league action boosted by a creditable display after they more than made a fist of it against top-flight opposition.
Their visit to Wearside came at precisely the wrong time for Poyet and his players with the first leg of their Capital One Cup semi-final against Manchester United scheduled for Tuesday evening, and the manager made seven changes as he attempted to manage his scarce resources.
But the Uruguayan returned to the dressing room at half-time with many questions remaining unanswered.
Poyet chose to field midfielder Ki Sung-yueng alongside Wes Brown on his return from suspension in central defence, while there was another start for striker Ji Dong-won on the left wing.
But it was Craig Gardner, prompted by midfield colleagues Johnson and Sebastian Larsson, who carried the biggest threat for the home side in the early stages with a series of long-range efforts, although only one of them forced visiting goalkeeper Greg Fleming to make a save.
Fleming did, however, distinguish himself with a fine one-on-one block to deny Jozy Altidore after he had been played in by Johnson with 17 minutes gone, and the United States international later scuffed another effort harmlessly wide from close range.
In the meantime, Prince Buaben had tested Sunderland goalkeeper Vito Mannone at his near post with the League One side still well in the game.
The visitors eventually fell behind 12 minutes before the break as Johnson produced a piece of real quality after Gardner had been tripped by David Amoo 25 yards out.
His curling free-kick was too well placed for Fleming and sailed inside the post to give the Black Cats the advantage they craved.
However, the lead lasted just nine minutes as Amoo atoned for his part in the opening goal by sending in a cross which had Mannone flapping helplessly and Robson latched on to the loose ball to smash it home off the underside of the bar despite Brown's best efforts on the line.
Mannone was grateful to collect James Berrett's deflected injury-time effort as the Cumbrians sensed an opportunity.
But the Black Cats restored their lead within five minutes of the restart when Johnson played Larsson's cross first time back across goal and Carlisle skipper O'Hanlon sliced his attempted clearance past Fleming and into the net.
The visitors immediately embarked upon the search for a second equaliser with full-back Brad Potts getting forward well in support of Amoo down the right, but with Sunderland starting to retain possession their hopes dwindled.
Ki and Ji departed for Phil Bardsley and senior debutant Duncan Watmore with 27 minutes remaining, and the former Altrincham striker almost made his mark within five minutes when he fed Altidore and then ran on to the American's back-heeled pass before forcing another good save from Fleming.
The Black Cats assumed complete control as time ran down with Watmore providing fresh legs and an enterprising outlook, and the visitors found themselves having to defend ever deeper as time ran down.
Ba, an 80th-minute replacement for Larsson, made sure in the final minute of normal time with a helping hand from Altidore to complete a solid afternoon's work.
- Sports & Recreation
- Adam Johnson
- Matty Robson
- Sebastian Larsson
- Greg Fleming