The Rams, who won at Arsenal in the previous round, went ahead in the first half but Barmby turned things around with a perfectly-placed shot and an 80th-minute penalty that was contested by the visitors, to see the holders safely through.
Paul McGuinness handed defender Nicolas Ioannou a full debut with Gyliano van Velzen deployed as a lone attacker. After a scrappy opening, Charni Ekangamene could only produce a wayward shot after a thrilling run by Barmby, who tricked his way past three opponents down the left.
Moments after Luke McCullough was fortunate not to concede a penalty, justice was perhaps done when the offside flag stayed down after a Will Hughes ball into the box and Kane Richards was allowed to turn and lash home with his left foot.
Only a brave stop by Liam Jacob thwarted Mason Bennett seconds later but the Reds regained some composure with Adnan Januzaj combining with Mats Daehli to force Mats Morch to pat away and the Norwegian keeper comfortably fielded a tame shot by van Velzen.
Jacob saved well from Joshua Lelan and Bennett while a double shout for handball against Ioannou and Tyler Blacket inside the box was survived. However, United carved out chances too with Morch clutching a Daehli cross-cum-shot and van Velzen unable to convert a Blackett cross.
Barmby looked the brightest spark and it was no surprise when he grabbed the equaliser, running at the defence and seeing his shot loop up to Morch. The keeper, under pressure, could only parry the ball back to the forward, who took it past the Norwegian shot-stopper and somehow found