The Premier League's bottom club needed a penalty shootout to see off Championship strugglers Sheffield Wednesday after 210 minutes of football.
Kenny Miller, Eddie Lewis, Robert Earnshaw and Fagan all converted their spot-kicks, while goalkeeper Lewis Price saved from Glenn Whelan and Kenny Lunt.
Scotland striker Miller had forced the shoot-out after cancelling out Steve Watson's opener at the start of the second half.
Derby, without a win in any competition for 18 matches, tore out of the blocks with Lewin Nyatanga, Miller, Gary Teale and Andy Todd all being denied by Lee Grant.
But just as it has been in the country's top-flight, good fortune was not with Derby and they fell behind Wednesday's first attack in the first half.
Jermaine Johnson crossed from the right and veteran Watson diverted the ball passed Price from 15 yards.
Derby's confidence visibly sagged and the home side could have had the tie wrapped up by half-time.
Wade Small on the left and Johnson on the right terrorised the Derby defence and both should have doubled Wednesday's lead.
Derby were really struggling to clear their lines with Claude Davis and Dean Leacock struggling with their confidence after being tormented by the Premier League's top strikers in recent weeks.
But whatever Paul Jewell said at half-time worked as Derby levelled two minutes after the break.
Stephen Pearson played in Miller and he neatly drove the ball home from 18 yards.
Wednesday substitute Leon Clarke and former Rams frontman Deon Burton both had chances to win the game late on.
At the other end, Miller flung himself at a Lewis cross but could only divert the ball over the crossbar.
In extra time, Clarke twice tested Price who was equal to it.
Jewell threw on Earnshaw in the second period and he came close to winning the game outright when his shot from six yards was deflected wide.
But the manager's first win as Pride Park boss was not long in coming as Derby prevailed in the shootout.