My gosh! All those replies with different approaches, and nobody (so far) seems to have tried using a hash <update> kvale's was the only one to use a hash -- it was so short, I missed it, and wrote it again myself </update>:

#!/usr/bin/perl

use strict;

my %chosen; # you don't need a 10-element array to draw from

while ( keys %chosen < 3 ) {
    my $pick = int( rand(10)) + 1;
    $chosen{$pick} = undef;
}

print join " ", keys %chosen, "\n";

# (you could "sort {$a<=>$b) keys %chosen", if that matters)
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This sort of problem does not merit a lot of optimization, but I think the extra iterations that might be needed on occasion to avoid repeat picks would, on balance, take less time than various amounts of array splicing, shuffling, etc...