Rivers also found the net on his debut for Bishop's Cleeve against Cinderford Town in December 2013.
Yate were 4-2 down on Saturday when they were awarded a free-kick just inside their own half.
He told Gloucestershire Live "I'd known from the first half how hard it was to judge long balls in the wind," Rivers, who played outfield until his early teens, said.
"I just decided to hit it goalwards and see what the wind did with it and luckily it went straight in.
"I didn't really celebrate as we were still a goal down with 12 minutes left, so we just ran back to restart."
The ball did not bounce on the way in, sailing into the net past Cirencester's stand-in goalkeeper Arie Peursum, promoted from the development team in the absence of number one Mat Wieczorek.
"Neither team was really happy with the result," Rivers said.
"But I have really enjoyed my football since signing for Yate last December."
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