One detail that I forgot to mention, and I edited the original post, is that I am working on Windows with Strawberry Perl. I had some trouble integrating the command into my script and could not store the output into a variable as it just went to stdout due to the "print $1 if" portion. I tried to redirect stdout, and not using "print" at all, but I ended up creating a temp file using this code to store the resulting number in a variable:

use IO::Handle;

# Extract the Payee ID from the file and store it into a temporary fil
+e
my $payeeid;
system( q(perl -ne "print $1 if /INC\*XX\*(\d+)/" cr835.txt > payee.tx
+t) );

# Define a file variable and read the contents to a variable
my $file;
$file = path("./payee.txt");
my $payeeid = $file->slurp;

# Define the date and time for use in the new file name
use constant DATETIME => strftime("%Y-%m-%d_%H-%M", localtime);

# Define the new file name
my $file = "CR835.txt";
my $newfilename = "CR835-" . $payeeid . "-processed_" . DATETIME . ".t
+xt";

# Rename the file
move $file, $newfilename;
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Is there a better way to store the result into the variable?

Yes - I showed how to do this without calling an extra perl process, it'll give you much more power and control.