your code is certainly correct (even if it doesn´t use strict;), but aatlamaz states that he couldn't handle newline problem. That leads me to the assumption, that at least one of the files has no newline at its last line. Thus it must be added.
So i changed above code to handle that (while also adding argument handling and support for more than two files):

use strict;

my $outfile = pop @ARGV   or die "No outfile!\n";

my @infiles = @ARGV       or die "No infile(s)!\n";

open OUTFILE, ">$outfile" or die "Cannot open $outfile!\n";

for ( @infiles )
{
        open (INFILE, $_) or die "Can't open $_!\n";

        while ( <INFILE> )
        {
                chomp;
                print OUTFILE "$_\n";
        }

        close INFILE;
}

close OUTFILE;
[download]

The script can be called like this:

shell: perl script.pl file1 file2 fileN outfile
[download]