I think it is clearer and less error-prone to open the file twice (input, then clobber and output) instead of opening once in read-and-write mode.
Working, tested code:

#!perl
use strict;
use warnings;

my $filename = '/tmp/test.index';

my $index = 0;
if ( -e $filename ) {
    open my $input_fh, '<', $filename
        or die "Failed to open '$filename' for input: $!";
    my $line = <$input_fh>;
    if ( not eof $input_fh ) {
        die "File '$filename' has more than one line! Dying to avoid c
+lobbering unexpected contents.";
    }
    close $input_fh or warn;
    chomp $line;

    $index = $line;
}

$index++;

open my $output_fh, '>', $filename
    or die "Failed to open (and clobber) '$filename' for output: $!";
print {$output_fh} $index, "\n";
close $output_fh or warn;
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