You have a couple of suggestions already, plus the explanation for why $1 and $2 survive successive loop iterations. Here is how I would modify the code:

use 5.010;
use autodie;

open my $fh, '<', 'text.txt';

while (<$fh>) {
    my @words = split /\s/;
    say "@words[0..1]" if @words >= 2;
}
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Note the use of autodie to avoid having to do explicit error checking on open or reads. Also note the use of three-argument open, which is an important security best-practice. Not necessary in your example, since your filename is a literal, but it's a good habit to get into.

It looks like you're really just splitting words on whitespace, so split seemed more natural and expressive to me. Maybe this was a contrived example to show the regex behaviour, and your real code really needs the regex, but for what's in front of me, split would be my choice.

Finally, I like say, but you can of course use print, and drop the 5.010 requirement if you are going for maximum backward compatibility.

Edit: My actual final word is, thanks for teaching your kid some Perl!