After taking a first half lead through Steed Malbranque at the Stadium of Light, Sunderland struggled to kill off their Blue Square Premier opponents in freezing Wearside temperatures.
But after Barrow, backed by more than 7,500 fans, had threatened to pull level on numerous occasions in the first half, they were outclassed after the restart when Campbell finally got going.
The former Manchester United striker, yet to score in the league for the Black Cats, finally got going and was the main reason why Sunderland took control.
Seven minutes after the restart he dived to head in his first from Daryl Murphy's centre and then he added his second five minutes later, this time flicking Phil Bardsley's cross beyond goalkeeper Tim Deasy.
SUNDERLAND (4-4-2): Fulop; Bardsley, Cana, Da Silva, McCartney (Liddle 82); MALBRANQUE, Meyler, Henderson, Murphy (Healy 77); Campbell, Bent (Noble 72). Subs: Carson, Anderson, Reed, O'Donovan.
BARROW (4-4-2): DEASY; Spender, Bolland, Pearson, Jelleyman; Goodfellow, Hulbert, Bond, Rutherford (Rutherford 56); Blundell (Rothery 72), Walker. Subs: Jones, Boyd, Sheridan, Bayliss, Tomlinson.