You need to reopen them to somewhere, e.g. something like

...

sub background_run {
    my @cmd = @_;
    
    if (my $pid = fork) {
        # parent
        # (...)
        
    } elsif (defined $pid) {      
        # child
        
        # This redirects both STDOUT/STDERR to the same logfile.
        # Instead of appending (">>"), you could of course also
        # create the file anew every time (">"), or create two 
        # separate logfiles, or whatever...

        my $logfile = "/tmp/bgrun.log";
        # re-open STDERR/STDOUT (closes first implicitly)
        # (if this fails, the error msg will end up in Apache's error_
+log)
        open STDERR, ">>", $logfile or die "Couldn't open $logfile: $!
+";
        open STDOUT, ">>", $logfile or die "Couldn't open $logfile: $!
+";

        # optional
        # my $app_path = "/path/to/dir/to/run/long-running-cgi/in";
        # chdir $app_path or die "Couldn't chdir() to $app_path: $!";

        exec @cmd;
        die "Couldn't exec '@cmd': $!";
        
    } else {
        die "Couldn't fork: $!";
    }
}

...
# files (perl, sleep.pl) must be found - in case of doubt,
# use absolute paths...
background_run("perl", "./sleep.pl");
...
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