You are correct that you need to run a daemon, but I don't think there should be any problem with daemonizing in the CGI before you exec(). For example,

#!/usr/bin/perl -w
use CGI qw( header );
use strict;
use POSIX qw( setsid );

my $pid;

# fork
die "cannot fork: $!" unless defined ( $pid = fork() );

# if parent
if ( $pid ) {
    # reply to user
    print header(), "Process forked.";
} else {
# we're the child
    # so we don't prevent filesystems from being unmounted
    chdir '/' or die "Can't chdir to /: $!";

    # redirect IO, or we'll be sucky and make the CGI hang
    open STDIN, '/dev/null' or die "Can't read /dev/null: $!";
    open STDOUT, '>/dev/null' or die "Can't write to /dev/null: $!";

    # thank you, POSIX ( I'd have hated to call ioctl() myself! ;-)
    setsid or die "Can't start a new session: $!";

    # lets wait until after setsid to redirect STDERR
    open STDERR, '>&STDOUT' or die "Can't dup stdout: $!";

# run our background process    
    exec("/path/process");
}
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