Steve Bruce's Hull side conceded two first-half goals but hung on to secure a place in the May 28 final at Wembley, which will bring a money-spinning spot in the Premier League worth up to 170 million pounds for the winners.
"We never do things easy but we have to give credit to Derby who on the night put up a hell of a show. We had to show resilience which saw us through in the end," said Bruce after his 200th game in charge of the Tigers.
"Over the two legs we have just shaded them. Talk about two extremes... It was a nervous, edgy night. We had to defend for an hour and we have done that to go through."
Derby opened the scoring in the seventh minute when Scotland striker Johnny Russell poked home from close range after his first effort was blocked following Chris Martin's downward header at the far post.
The visitors made it 2-0 on the night, and 3-2 on aggregate, in the 36th when Andrew Robertson put through his own goal from close range after a low left-wing cross from Andreas Weimann was diverted by Hull defender Curtis Davies.
But despite some nervy moments for Hull, whose fans were left biting their nails as the clock ran down, they managed to hold on to their advantage to book a place in the final.
Hull finished fourth in the final table, five points ahead of fifth-placed Derby, and will face a Wednesday side who completed a 3-1 aggregate victory over Brighton & Hove Albion with a 1-1 draw on the south coast on Monday.
"We'll have to be better (in the final) and we will because we couldn't be much worse. A big Yorkshire derby down at Wembley -- let's look forward to the occasion," Bruce added.
A Hull win would take them back to the top flight at the first attempt, which Burnley managed by finishing top ahead of Middlesbrough, who took the second automatic promotion spot.
(Writing by Ken Ferris, editing by Ed Osmond; Editing by Ed Osmond)
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