fishbot_v2 has my vote for best solution within the constraints. However, if you were doing this repeatedly, using index might be an interesting approach. I've coded fishbot_v2's solution into the regex sub below. I did several runs and pasted a pretty typical result.

use Benchmark qw/:all/;

sub regex {
    return ($_[1] =~ m/^$_[0]/) ? 1 : 0;
}

sub index_func {
    return (index($_[1], $_[0]) == 0) ? 1 : 0;
}

cmpthese ( 500000, {
    'regex match' => sub { regex('abbr', 'abbreviation') or die },
    'index match' => sub { index_func('abbr', 'abbreviation') or die }
+,
    'regex miss'  => sub { regex('anna', 'abbreviation') and die },
    'index miss'  => sub { index_func('anna', 'abbreviation') and die 
+},
});

__END__
                 Rate regex match  regex miss  index miss index match
regex match  680272/s          --        -13%        -36%        -38%
regex miss   781250/s         15%          --        -27%        -29%
index miss  1068376/s         57%         37%          --         -3%
index match 1101322/s         62%         41%          3%          --
[download]

As you can see, using index for this is much faster, regardless of whether you are more likely to match or miss.