The 19-year-old midfielder and the £6million striker had the game sewn up within 23 first-half minutes as a much-changed City side offered little resistance in front of a crowd of just 20,576 at the Stadium of Light.
They might have won even more comfortably had Maik Taylor not denied Kenwyne Jones with his legs, or had defender John Mensah's 57th-minute strike not been chalked off for a foul.
Black Cats manager and former Birmingham boss Steve Bruce, who had bemoaned his team's inability to produce a 90-minute performance in the wake of their 3-1 Barclays Premier League defeat at Burnley on Saturday, looked on as they did not even have to turn it on for 45 minutes to secure their place in the hat.
The respective priorities of the two managers became abundantly clear when the teams were announced.
Bruce, who believes the Carling Cup perhaps represents his side's best chance of winning something this season, made just four changes to the team which started at Burnley on Saturday and even then those changes hardly weakened the Black Cats line-up.
In came Jones, for whom Tottenham were prepared to pay £15million in January, seasoned internationals Mensah and Paulo da Silva, and promising youngster Henderson.
By contrast, opposite number Alex McLeish, whose main focus is simply Premier League survival, rested seven of the men who played at Hull City at the weekend.
Brushed asideThe resulting disparity was if not inevitable, at least hardly surprising, although the ease with which City were brushed aside during the opening exchanges would have been a matter of some concern for the Scot.
Sunderland were ahead within four minutes, Henderson sweeping home his first senior goal for the club from Jones' square ball to set the tone for a blitz which the visitors found irresistible.
Birmingham's plight would have deepened on 15 minutes had Taylor not blocked Jones' stabbed effort from just seven yards with his legs, but the respite was short-lived.
It was 2-0 with 23 minutes gone when Campbell, who had already seen a close-range header deflected over by defender Giovanny Espinoza, made amends, beating Taylor to Andy Reid's inviting cross to glance the ball home for his first Sunderland goal.
Full-back Kieran Richardson was finding plenty of space down the left to get beyond Reid, and he did so to good effect two minutes before the break to float a cross over the goalkeeper which skimmed the crossbar and ran safe.
For their part, City offered little of any note as an attacking force, although Sunderland keeper Craig Gordon was sent scurrying across his line with 19 minutes gone as Gary McSheffrey's well-struck drive flew just wide.
To their credit, the visitors rallied briefly immediately after the restart, but it did not last, and they might have fallen further behind in a flurry of activity in the 53rd minute.
Good blockFirst former Middlesbrough defender Stuart Parnaby had to get in a good block to keep out Reid's driven effort, and then Taylor had to pluck Lorik Cana's curling shot out of his top corner.
Mensah thought he had wrapped up the win when he stabbed home from two yards out with 57 minutes gone, but referee Mike Dean blew for a foul in the melee from which the ball had dropped at the defender's feet.
The visitors almost took advantage three minutes later when Jones cleared a McSheffrey corner only as far as Garry O'Connor on the edge of the box, but his piledriver was blocked by a combination of Henderson and Mensah.
Jones could have sealed the win with 20 minutes remaining when he collected Reid's pass 20 yards out, but he lifted his shot high over the crossbar.
O'Connor was equally wasteful at the other end seconds later after Michael Turner had blocked his initial effort.
Reid went close six minutes from time, but the damage had already been done.