Where to start? First of all, no one "balances" a bike, as in sitting perfectly still and the bike stays upright. What we do is continually fall left and right while steering the bike back underneath us. At speed, the spinning gyros, (the wheels), heavily resist this falling, so our corrections are minute and we enjoy the illusion of balancing. Slow down to a crawl, however, and the process becomes immediately evident.

Now, track stands. Again, no one balances. The track rider cocks his wheel up the track, and the road rider cocks his into the crown of the road. This allows the bike to move left and right. Push the pedal and go up the slope, release pressure and go back down. Left and right.

Note that the track rider employs this technique even though he has the ability to go backwards by pedaling backwards. But forward and backwards won't keep you upright. You must go left and right, hence the cocking of the wheel into the slope even on a fixed gear bike.

Tandems, as you've witnessed, can do the same thing. Cock the wheel into the crown of the road and use pressure/no pressure on the pedals to move incrementally left and right. As PMK and others mentioned, it requires:
A. Stoker confidence
B. Stoker's ability to sit perfectly neutral
C. Captains ability to track stand a road bike in the first place, and
D. Practice, practice, practice and patience, patience, patience.

Suggestions:
Drop the stokers seat down to where she can reach the ground flat-footed. Someone else mentioned using sneakers, not cleated shoes. These things will help inspire the willingness to try. Find a soft-looking, grassy incline to practice on, and have a go at it. Remember, the stoker will ALWAYS get nervous and put her foot down ahead of you. Expect it, discuss it and accept it. It's YOUR job to overcome her fear, (i.e, instill confidence in your ability), not hers.

You'll be wobbly at first, and your corrections will probably be large and erratic. Standing on the pedals is a good technique for starters, as it allows you to add some arm muscle into the act. You'll find yourself doing some amount of "throwing" the bike up and down the incline to keep your balance, but you'll slowly settle in and can sit down.

You'll know you're REALLY good when you're out on the road again with the Saturday group, you get into a tandem track stand at a light, you sit up and take your hands OFF the bars, and your stoker doesn't bat an eye.