Gus Poyet's side, who sit bottom in England's top flight, made the most of a break from their league woes to book a home clash with Kidderminster Harriers or Peterborough United, but were made to work hard against the League One outfit.
After dominating the early stages of the first half, the hosts took the lead 34 minutes in through a stunning free-kick from Adam Johnson.
David Amoo fouled Craig Gardner on the edge of the area and Johnson curled the ball home left-footed, but Sunderland's lead was shortlived.
Matty Robson was the man on the scoresheet for the visitors two minutes before half-time, pouncing on a loose ball to score after the home defence failed to clear Amoo's cross.
Sunderland restored their lead soon after the restart thanks to a large slice of luck, with the unfortunate Sean O'Hanlon turning Johnson's effort over goalkeeper Greg Fleming, before a debut goal from substitute El-Hadji Ba finished off Carlisle a minute from time.
Poyet brought Gardner and Wes Brown back into his starting XI, although captain John O'Shea remained absent due to a persistent shoulder problem.
Former Sunderland midfielder Graham Kavanagh made three changes to Carlisle's team, handing starting berths to Bradley Potts, Lewis Guy and O'Hanlon.
Jozy Altidore found himself one-on-one with Fleming after 16 minutes courtesy of a throughball from Johnson, but Carlisle's keeper came out on top.
Sunderland maintained the pressure and Gardner was next to threaten from long range following a dangerous corner from the home side.
Vito Mannone was called into action for the first time on 21 minutes when he tipped Prince Buaben's driven strike round the post after it evaded a clutch of Sunderland defenders.
Altidore then failed to get sufficient contact on a teasing Ondrej Celustka cross before Johnson's free-kick eventually handed Sunderland the lead their early dominance warranted.
The winger's third goal of the season, all of which have come at the Stadium of Light, looked set to put Sunderland in front at the interval only for Robson to draw Carlisle level.
Sunderland retook the lead four minutes into the second half when Johnson's volley was diverted into the Carlisle goal courtesy of a deflection off the covering O'Hanlon.
Having already conceded one lead, Poyet's men appeared determined to gain a two-goal cushion but struggled to convert their superior possession into clear-cut chances.
Debutant Duncan Watmore showcased his eagerness to impress after being introduced a minute after the hour, the 19-year-old forcing a smart save from Fleming after neat interplay with Altidore.
Carlisle continued to pose a threat on the counter, but Ba completed the scoring following Altidore's lay-off as Sunderland progressed to the next round of a competition they last won in 1973.
