Following a first leg in which Derby took a narrow 1-0...
Trailing 1-0 from the game at Pride Park, goals from Ryan Sessegnon and Denis Odoi put the hosts through.
The Rams struggled to create opportunities on the night but they were comfortable in the first-half as they dealt with consistent Fulham attacks.
Derby will now spend an 11th-successive season in the second tier despite reaching the play-offs in three of the past five campaigns.
However, two minutes after the break, Fulham opened the scoring on the night and levelled the tie, Sessegnon getting the goal.
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The 17-year-old provided the telling assist 19 minutes later, bending a corner onto the head of Odoi, who glanced the ball over Carson and into the left-hand corner of the net to keep Fulham's promotion hopes alive for one more match.
Fulham head to Wembley for the first time since they lost the FA Cup final to West Ham in 1975.
Sessegnon said afterwards: 'We pushed and pushed and should have scored early in the first half. "When we had the chance to pass the ball, we looked edgy and we didn't handle the ball well enough".
Derby boss Gary Rowett felt his team had been too nervous.
"To get to half-time at 0-0, we felt that we had to step out a bit more and be more proactive".
"We have to assume that they will play with good composure and quality, and we made some really poor decisions during the game".