Presuming your aliases are defined in .aliases (and in bourne shell compatible syntax), this should work:

my $run_cmd = 'l';

open(COM, ". .aliases\n$run_cmd 2>&1 |") || die "Could not execute com
+mand\n";
...
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Note that there must be a newline (not a semicolon) after sourcing the .aliases file, because aliases cannot be used on the same line that they are defined.

P.S.: not all shells will by default expand aliases, unless in interactive mode. For example, sh does, while bash does not (even if sh is just a symlink to bash), but you can explicitly enable the feature via shopt -s expand_aliases.  Hence, the above example only works because perl calls /bin/sh -c behind the scenes (at least typically — check with perl -V:sh).