An individual goal by 20-year-old Uruguayan Nicolas Lopez, only two minutes after coming on as a substitute, condemned Milan to the inevitable chorus of jeers from a sparse San Siro crowd after their quarter-final defeat.
Milan had given Seedorf a winning debut by beating Hellas Verona 1-0 in Serie A on Sunday and made a flying start to the tie when Mario Balotelli tapped in from close range after six minutes.
Milan failed to build on their early goal and Udinese levelled with a penalty converted by Colombian forward Luis Muriel four minutes before halftime following Urby Emanuelson's clumsy tackle on Silvan Widmer.
Udinese looked far more dangerous after halftime and were only prevented from going ahead by a dreadful double miss midway through the second half.
Substitute Maxi Pereira scuffed his shot in front of a gaping goal, then managed to scramble it back to Muriel who also miscued from a good scoring position.
Lopez showed them how it should be done when he collected the ball just inside the Milan half, surged forward and fired his shot into the corner.
Milan, 11th in Serie A, had no answer and trudged off after writing another chapter in a dismal season.