I had the exact same issue as you, except my leak was on the driver'* side. I did what Peter suggested, had the glass removed and left it out for 4-5 days in my garage. I did the necessary repairs and then had the glass company come back out to reinstall it. It doesn't leak a drip now.
Here'* what I had to work with when they removed the glass:
Yours likely looks the same underneath. I got carried away with the repair and didn't take any pics, but what I did was wire wheel the rusted areas. That got rid of the debris and rust. Then I used a pneumatic grinder to grind down to bare metal. After that I masked off all potential overspray areas and put 4-5 coats of flat black Rust O Leam aresol paint on the affected areas. 2 days later the glass guys came back and reinstalled the windshield. Like I said, I haven't had any issues.
If you rig it with caulk or filler at the trim, you're only delaying when that leak will resurface.
This is going to Detailing and Appearance...