In a game played in testing and boggy conditions, it was Snainton who sealed the three points with just 10 minutes left on the clock.
Regan Hewitt's corner was only partially cleared and it fell kindly for centre-back Ross Triffitt to fire in the only goal of the game on the half-volley.
Boss Andy Holt was delighted with his side's efforts in tough conditions.
"There weren't many chances in the game, but maybe we adapted to the conditions in the second half and just about edged it," he said.
"It was difficult to get going but the players did well and deserved the win.
"The whole team deserves a mention but Andy Worgan, Sam Turner, Sam Cooper and Ryan Hunter stood out.
"It keeps us in the hunt, but we are missing Ryan Collins, Rob Holt and Liam Cummings through long-term injuries so we will have to work hard and get a bit of luck if we are to try and win promotion."
Cayton player-boss Shaun Rennison wasn't impressed with the condition of the pitch.
"It was a scrappy game that wasn't helped at all by the pitch," he said.
"There weren't a lot of chances in the game. Whoever took their chance in the end was going to take the win and they did that.
"We won't be able to win the league now, which is what we wanted to do, but I believe we'll still get promoted if the lads keep turning up and working hard."
Rennison singled out defenders Jake McAleese and Adam Robertshaw as his side's standout performers.
MandX Partners Ltd were matchball sponsor.
Flamborough conceded their home game against Seamer in Division One earlier in the week as they were unable to raise a side.
With the rest of Saturday afternoon's schedule wiped out by the wet weather in the area, the only other game to go ahead was Friday night's Division Two clash between Itis Itis Rovers and Falsgrave.
Player-boss Mikey Barker led the Rovers charge as they cruised to a 10-1 win that keeps them on course for the Division Two title and back-to-back promotions.
Barker hit a treble with Sean Bloom and man of the match Neil Forsyth adding doubles.
Liam Buglass, Sam Pickard and Callum Myers also found the back of the net in a one-sided encounter.