The Rams lost in the last minute of the 2014 Wembley final and missed out on a top-six spot on the final day of the season a year later.
A tepid display in the home leg of their semi-final, against a Hull team over whom they completed a league double in 2015-16, left them with little chance of reaching the final.
For all of their early pressure, Derby were unable to test Hull goalkeeper Eldin Jakupovic – a late addition to the starting line-up after Allan McGregor suffered a back injury in training on Thursday.
Bradley Johnson gave away possession in the build-up to Hull’s first goal, while the second came with a degree of poor fortune as goalkeeper Scott Carson was powerless to keep out Odubajo’s hugely deflected shot five minutes before half-time.
Jacob Butterfield’s tame shot from distance was Derby’s first effort on target – with 10 minutes remaining – and was greeted by ironic cheers from the home supporters, who booed loudly as Robertson’s excellent finish almost put the outcome of the semi-final beyond doubt.
Hernandez hits form
Hull’s top scorer Hernandez scored his 22nd goal of the season to put his side ahead in their semi-final first leg.
The Uruguay international went through a lean spell, going nine appearances without a goal between February and April, but has returned to form in recent weeks.
Four goals in his last five outings of the regular season helped Hull finish fourth in the table, and the 25-year-old striker netted for his fourth game in succession to take the Tigers closer to an immediate return to the Premier League following relegation last term.
Steve Claridge, ex-Leicester striker, on BBC Radio 5 live:
“I am absolutely stunned. I thought this was going to be a close tie. Everything Hull have done has worked perfectly; everything Derby have done has just not worked.
“Hull are the team that have looked the more dangerous going forward. At 3-0 you fear for Derby in the second leg; there is nothing for them to play for.”
Derby head coach Darren Wassall: “It’s not the result we wanted and certainly not the result we expected, considering how well we’ve played at home over the last few weeks.
“We’ve got two choices – we can either throw the towel in now, which is not what we’re going to do, or we can regroup, show some character and resilience and dust ourselves down and go again on Tuesday.
“That is what we’re going to do because we can’t end the season on a performance like that.”
Hull manager Steve Bruce: “In big games you want your big players to turn up and I thought today our experience certainly showed.
“It was a really, really good performance from start to finish – which in a big game is great to see.
“We know we’re capable of that but what we have got in our locker is complacency, and we’ve got to guard against that on Tuesday.”