Trinity were leading 2-0 at fellow strugglers AFC Telford United but the Bucks fought back to win and condemn Gainsborough to the drop.
It is the first time the club have suffered relegation in their 145-year history and they will kick off next season in the Northern Premier League.
Sinnott has overseen an upturn in results since he took charge in mid-February but they haven’t been enough.
“It’s massively disappointing but to be 2-0 up and lose it - that’s a snapshot of the season,” boss Sinnott said.
“I’ve only been here for two months but that bit of fragility has been there all season.
“I feel sorry for the supporters who have made the journey down but we have to be realistic - we’re a relegated team.
“It’s not a nice thing to digest but now begins the work for next season. I’ve been given the inclination that I’ll be here next season and I haven't heard anything different.
“I’m looking forward to the job in hand. There will be change because when things like this happen, there has to be change.”
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