Midfielder Flynn put four-time Cup winners Sheffield ahead in the 65th minute, when he snuck in at the far post to nudge home a left-wing cross from Jose Baxter.
The home side doubled their lead 1 minute, 42 seconds later when defender Brayford’s low, long-range effort took a cruel deflection off the visitors’ Richard Wood and rolled into the net.
At KC Stadium, Hull reached the semi-finals for the first time since 1930 after striker Sone Aluko missed a first-half penalty against Sunderland.
A game of poor quality was brought to life in the 68th minute when defender Curtis Davies headed the opener.
Four minutes later, David Meyler was too quick and strong for Lee Cattermole, who allowed the midfielder to burst past him at the halfway line and slip the ball past Oscar Ustari for the second.
Meyler celebrated by running toward the sideline and headbutting the corner flag in an apparent reference to an incident last week in which Newcastle United boss Alan Pardew was sent off for aiming a headbutt at the Irishman during a Premier League game.
Five minutes later, Cattermole was again the Sunderland culprit, inexplicably aiming a backpass toward his ’keeper, but playing the ball straight to Matty Fryatt, who then produced the third.