Back row forward Grieve scored the only try of the match after 10 minutes
Bristol's play-off campaign got off to a winning start but they struggled to overcome a stubborn and spirited Cornish Pirates side.
After Pirates opened the scoring, Paul Hull's side led 10-3 after 10 minutes thanks to an Adrian Jarvis penalty and Iain Grieve's converted try.
A 10-6 half-time lead became 10-9 when Pirates full-back Robert Cook kicked his side's third penalty.
But three second-half Jarvis penalties saw Bristol home to a narrow victory.
The Pirates got off to a flying start in the first minute as fly-half Johnny Bentley broke through the Bristol defence and was brought to ground 10 metres short of the try line.
The home side were penalised for not rolling away from the tackle and full-back Cook slotted over.
Bristol were soon level through Jarvis's penalty, then scored the only try of the match from flanker Grieve.
A good clearance kick from Bristol's Luke Arscott gave the Pirates a line-out in their own 22 but Bristol stole the ball and Grieve touched down after spinning off a good rolling maul.
It seemed that Bristol would kick on from there but they lack any real fluidity in their play, kicking away a lot of possession.
The visitors put together a good few passages of play in the Bristol half, allowing Cook to narrow the deficit to just four points with his second penalty after 24 minutes.
The score then remained 10-6 until the 52nd minute when Cook slotted a third penalty for the Pirates.
The second-half was an uninspiring affair, with neither team able to keep hold of possession.
Bristol were indebted to fly-half Jarvis who had a 100% record in front of the posts. And he kept his side's noses in front with two penalties of his own.
With just five minutes left, after a fourth Cook penalty, Bristol were leading 16-12. And they had to survive some nervy moments defending their own try line before another stolen ball at the Pirates line-out allowed them to clear their lines.
And Jarvis added a fourth penalty in the final seconds to bring relief to many of the 5,892 watching.
Bristol head coach Paul Hull told BBC Radio Bristol:
"There's a bit of disappointment because we didn't play as well as we can.
"I think there were a few nerves within the team which is surprising as we've been playing for a long time now."
"It's pressure rugby at the moment and this is the first of hopefully a few games played in this environment."
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