It should work, and it does work for me.

>type a.pl
` perl -e"print qq{stdout\\n}; warn qq{stderr\\n};" >stdout 2>stderr `

>perl a.pl

>type stdout
stdout

>type stderr
stderr
[download]

Just to make it more simplistic I tried redirecting STDERR to a file [...] ...which does not work (meaning I don't see anything within output.txt).

So where does it go? If nowhere, then you are either getting a message such as "Access is denied" on STDERR or the child doesn't send anything to STDERR.