I inherited an established garden in which a very sad-looking peony was struggling to grow right next to (practically from under) a rose. In the two growing seasons I've been at this house, the peony's few leaves were a yellow-green and it never sent up a flowering shoot. In May this year I moved the rose that was next to it (due to a color scheme I didn't agree with), and at the same time, I removed the peony. It looks as if the peony managed to grow from a bit of root was left after the previous owner tried to remove it and replace it with some roses, but perhaps it was too deep in the soil, or constantly cut back, or too malnourished to grow successfully. I potted up the bits of root, some with leaf still attached, hoping that I could get at least one plant. I've been keeping the potting compost moist but not water-logged and keeping the pots out of direct sunlight. Is this the right thing to do to save the peony, or is there a better method?
Peonies are often shipped by nurseries as root stock, and they can be propagated from toot cuttings (which you've found!), so there are quite a few nursery websites with pages about caring/planting peony root stock. Peonies are apparently also noted for their longevity, so I think you've got a good chance to keeping it going.
This article provides a more complete text on peony care and propagation (but written in a horrible script font - you may want to change your browser font settings).