Recursion is the trick:

use strict;
use warnings;

print "$_\n" for GenUniStrings(3, 10);

sub GenUniStrings {
    my ($numOnes, $strLen) = @_;
    my @strings;

    die "Number of ones can't be zero or negative" if $numOnes < 1;

    for my $prefixSize (0 .. $strLen - $numOnes) {
        my $prefix = '0' x $prefixSize;
        my $tailLen = $strLen - $prefixSize - 1;

        if ($numOnes == 1) {
            push @strings, $prefix . '1' . ('0' x $tailLen);
        } else {
            push @strings, map {$prefix . '1' . $_}
                GenUniStrings($numOnes - 1, $tailLen);
        }
    }

    return @strings;
}
[download]

Prints:

1110000000
1101000000
1100100000
1100010000
...till...
0000001110
0000001101
0000001011
0000000111
[download]

Update: minor code tidy