To me the secret to DRY (which is Do not Repeat Yourself -- I had to look that up) is to convert functionality to functions, encapsulate it in objects as methods that only execute in certain states, or to utilize some core language feature to reduce the code.

From your desire, "I need to ensure that a variable is decremented regardless of how an if clause is exited.." it reads like you want the decrement operation to happen automatically when something else occurs -- a form of encapsulation -- like this:

use strict;

my $g_var = 1_000;

sub special_dec {
    my $choice = shift;
    if  ($choice)
    {
        --$g_var;
    }
}

my $thing = 'dog';

if ( $thing eq 'cat')
{
    my $cntr = 0;
    while (++$cntr < 11) {
         last if (special_dec(900==900));
    }
}
elsif ($thing eq 'dog')
{
    my $cntr = 0;
    while (++$cntr < 11) {
         last if (special_dec(900==900));
    }

}
print $g_var;
1;
# of course prints 999
[download]

Now you don't have to think about manually decrementing or handling $g_var, something else does that for you.

Refactoring your solution may mean that you really need to write special operations that allow you to iterate over lists and handle certain internal values without the user of that object having to consciously know about them. I think in Perl there is a desire to try and solve a great many problems using language constructs and coding techniques when oftentimes the more obvious solution is to just push certain actions into functions or object methods instead. If you are doing Perl golf then disregard this post but your question doesn't read like that.