($a) cmp ($b) is comparing the full strings, not just the alphabetics, so perl never gets to the second comparison (which should be or'ed, BTW). You need something like

my @split_list = qw{erez[11] dana[22] dana[0] erez[10] erez[1] erez[0]
+ dana[10]};

my @new_list = sort {
    ($a =~ /(\w+)/)[0] cmp ($b =~ /(\w+)/)[0]
                       ||
    ($a =~ /(\d+)/)[0] <=> ($b =~ /(\d+)/)[0]
} @split_list;

print "$_\n" for @new_list;
__END__
dana[0]
dana[10]
dana[22]
erez[0]
erez[1]
erez[10]
erez[11]
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