The scoreline gave Heerenveen a 3-2 aggregate triumph and set up a playoff final with Vitesse Arnhem, who defeated PEC Zwolle 3-2 on aggregate earlier in the day.
Trailing 1-0 from the first leg, Feyenoord took the tie to extra time when Anass Achahbar netted four minutes before halftime.
The Rotterdam club were forced to play the extra period with 10 men when substitute Luke Wilkshire was forced off through injury and the home team had already made their three changes.
Heerenveen took advantage with Uth's double blow in five extra-time minutes, before Colin Kazim-Richards found an equaliser on the day, but there was too little time left for Feyenoord to salvage the tie.
It is a disappointing start to his tenure for rookie manager Giovanni van Bronckhorst, who took over Feyenoord for the playoffs after predecessor Fred Rutten left at the end of the regular season campaign.
He will continue next season.
Vitesse booked their place in the final with a 1-1 home draw against the 10 men of PEC Zwolle.
Burkina Faso midfielder Bertrand Traore had Vitesse ahead on 18 minutes when he netted his 14th of the league season.
Czech midfielder Tomas Necid equalised 11 minutes after halftime, but PEC could not find another to force the game to extra time and had defender Maikel van der Werff sent off late on after he collected a second yellow card.
(Reporting by Nick Said; Editing by Mark Meadows; mark.meadows@thomsonreuters.com; +44 20 7542 7933; Reuters Messaging:; mark.meadows.reuters.com@reuters.net)
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