Sheffield Wednesday 2 Hull City 4
Barmby tapped home after good work from Delroy Facey and then scored a stunning volley from 15 yards to give Hull a commanding lead, after Joey O'Brien and Michael Keane had traded early goals.
Wednesday swept into a ninth-minute lead. John Paul McGovern exploited space down the right, releasing Glenn Whelan whose cross was turned towards goal by Hull defender Marc Joseph.
Boaz Myhill produced a reflex save to tip the ball in the air but O'Brien celebrated his first game of a one-month loan deal from Bolton by following up with a close-range header.
Keane brought this electric derby level after 14 minutes, blasting home a penalty after Whelan was adjudged to have used his hand to block Damien Delaney's header.
Hull took the lead in the 22nd minute when Facey capitalised on an error by Lee Bullen to race through on goal. His cutback was scuffed by Keane but the ball ran to Barmby, who fired home.
Barmby showed his experience with a first-time volley two minutes before half-time, turning on Ryan France's flick to fire past David Lucas from 15 yards.
Patrick Collins had to make a last-ditch block to prevent Stuart Elliott making it four on the stroke of half-time.
Lucas made a great save from Elliott's close-range header after the re-start.
McGovern pulled a goal back for Wednesday nine minutes from time when he volleyed home after Paul Heckingbottom's long throw.
Tigers substitute Danny Allsopp made sure of the points two minutes later, breaking clear of the defence to slot the ball past Lucas.