Update: Otherwise, you can still make it shorter:

die+map{(Fizz)[$_%3].(Buzz)[$_%5]||$_}1..100
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Generally printing to STDERR is not considered equivalent to printing to STDOUT.

Update 2: As Sidhekin pointed out to me, the -l switch makes the newline problems go away.

But under common golfing rules -l does count as +3 strokes. Changing

$_ for
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to

$_,$/for
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adds two.