Blues went a goal in front after only three minutes through defender Peter Gilbert and Barrowman was unlucky not to double the lead when he outpaced the Sunderland defence but shot wide.
Blues had a scare just before half time when Capper rattled a shot against Clint Davies's crossbar but carried their lead through to the half-time break.
The home side pressurised the Blues defence more after the interval and Robert White equalised for the Black Cats on 58 minutes.
Davies then saved well from Sunderland substitute George with a quarter-of-an-hour left but, with just five minutes remaining, young striker Barrowman grabbed the winner for Blues.
The same result in the Premiership game at St. Andrew's on Saturday would please the Blues faithful, I'm sure!
BLUES: Davies, Capaldi, Luntala, Gilbert, Vickers, Swierczewski, Coly, Grondin, Kirovski, Barrowman, Motteram. Subs: Luckett, Doyle, Curtis, Beauchamp, Hipkiss.
SUNDERLAND: Turns, Wright, Capper, Dickman, Clark, Thome, Black, Whitley, White, Oster, Ryan. Subs: Peeters, Straker, Byrne, Flynn, George, Scott.
