There are two sides to this.
Writing a web front end
Generally speaking, you should use Plack/PGSI for this. It acts as a middleware layer that lets you run a Perl web application in a variety of ways (I usually go for FastCGI, other options include mod_perl and CGI).
You can use a framework such as Dancer or Catalyst (which use Plack under the hood), but they are probably overkill (depending on what the script does).
If you needs are very basic, you can use CGI without Plack (the CGI module makes this quite easy). CGI isn't very efficient, but it is very easy to configure. You won't notice the inefficiencies unless you are calling the script frequently or the script has a high startup time.
Accessing the existing script
The quick and dirty way would be for your web application to spawn a shell and run it in that. You could use backticks, system or exec or, if you want more control, IPC::Run.
The robust way would be to break out the business logic of the script into a module and rewrite the command line program to parse arguments and then involve the appropriate functions from the module. You then do the same thing for the web application (the difference being how you present the input and output).