The webserver should normally terminate the CGI script in case the browser prematurely closes the connection.  So, you could try putting your cleanup code in an END { ... } block, which should execute upon termination of the CGI script.

In case that doesn't work, you'll probably also have to set up a signal handler (most likely for SIGTERM, but I haven't checked¹) calling die or exit, which in turn will make the script terminate in a controlled fashion, so the END block executes.  (Details depend on the webserver, OS, etc.)

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¹ in a private msg Perlbotics noted (thanks!) that according to his observations, it would rather be SIGPIPE — which makes perfect sense, as the script would be trying to output to a broken pipe (presuming it hasn't already finished writing at the time...).