Run it through it? No.
I put my chiller in the boiling wort about 15 minutes before flameout. That is plenty of time to sanitize the outside of the chiller, which is the only part that comes in contact with the wort once it drops to a cool temperature.
Hook up your hoses or water lines before you put it in the boiling wort--and if you have any water left in the chiller, look out as it may be ejected as steam or boiling water.
You could, I suppose, clean the external parts of the chiller and then dunk in star san, but boiling it works as well. When I'm done chilling I remove the chiller, cover the kettle so nothing bad gets in, and immediately take it to a sink to rinse it off. I scrub off any hop residue that might be on the risers, rinse again, and that's that. The next time, it'll be sanitized by boiling--the boiling wort.