The visitors made the perfect start as Morgan MacRae powered through two defenders to run in from 25 metres out, Mike Cooke with a simple conversion.
Their opponents had two quickfire chances to reduce the arrears as a brace of long-range Tom Quinlan kicks from in front of the posts drifted wide right.
Rams then suffered a setback as captain Max Hayman was forced off with a trapped nerve in his back, but replacement Paddy Harris continued to help the away side dominate at scrum time.
Second rows Toby Hutchinson, the Clifton skipper, and Harry Stone were both sin-binned following a minor skirmish, the hosts beginning to dominate possession and territory with the wind at their backs.
The Rams defence, however, held up strong, and with no clear-cut chances at either end, went into the break with a 7-0 advantage.
Hayman returns, Rams extend lead
Hayman returned at the start of the second period, and his side extended their lead when Zach Clow ripped the ball near the Clifton line, Stone sending MacRae over on the left for his eighth of an increasingly productive start to the season.
Yet Clifton hit back immediately, claiming a box kick before scything through in midfield, young wing Ollie Harris finishing smartly in the left-hand corner.
Quinlan found his range with a terrific touchline conversion, and the game continued to ebb and flow with Rams twice penalised at attacking five-metre scrums.
Clifton level before Graham strikes
And in an increasingly fractious affair, they were made to pay for not taking their chances, the hosts level with 12 minutes remaining after a penalty on the away 22 was reversed.
The ball was kicked to touch, Tom Hatch mauling over for a try again added to by Quinlan for a level game.
Rams crossed the whitewash only for James Baker to be held up over the line, but a moment of real quality eventually proved decisive.
Mikey Duda made the initial break, and after replacement scrum-half Ollie Hodgson went back down the short side, some lovely hands from Clow sent Luke Graham away for his fifth try in three games this term.
There was still drama, though, Rams penalised for chat allowing their rivals to kick into the 22, but a breakdown penalty allowed Hayman to boot the ball off for a gritty triumph.
Rams return home for the final fixture in the opening block, Blackheath in town next Saturday (3pm).
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