Gareth Steenson outscored counterpart Adrian Jarvis three kicks to two
Exeter will take a narrow three-point lead into next Wednesday's Championship play-off final second leg after edging Bristol at Sandy Park.
The two sides were locked 3-3 at the break after Gareth Steenson and Adrian Jarvis swapped penalties.
Jarvis then booted Bristol ahead for the first time, only for Steenson to reply with two more kicks, the last of them just before the final whistle.
But Exeter had hoped to go back to the Memorial Stadium with a bigger cushion.
And Bristol now start as slight favourites for the second leg (Wednesday 26 May, 1945 BST).
In front of a full house at Sandy Park, Exeter enjoyed the better start, using fly-half Steenson's precision kicking to put themselves into Bristol's 22 on several occasions.
The visitors succumbed to the pressure in the eighth minute, giving away a penalty which allowed Steenson to put the Chiefs in front.
A fantastic game of rugby - Baxter
Bristol's first attacking chance came midway through the half, when a probing run from lively winger Tom Arscott enabled his brother Luke to kick ahead for Lee Robinson, who was only prevented a try by a desperate scramble from the Chiefs' defence that saw the ball go dead.
Buoyed by their close encounter with the Chiefs try-line, Bristol began to enjoy greater possession, turning the screw and forcing the home side onto the back foot.
Shortly before the break, Bristol came close again, only a superb try-saving tackle from Chiefs wing Paul McKenzie preventing Tom Arscott going over in the corner.
Bristol were eventually rewarded for their effort on the stroke of half-time when fly-half Jarvis put over a penalty of his own.
It is only half-time - Hull
The second-half saw Bristol come out the stronger, with their backs, led by the impressive Robinson, continuing to make inroads into Exeter's defence.
Paul Hull's men got their reward after 17 minutes when the Chiefs failed to withstand the pressure of several five-metre scrums and gave away a penalty which Jarvis duly slotted over.
Steenson pulled the Chiefs back onto level terms with a penalty after 26 minutes, but Bristol continued to have the run of play, and with just 10 minutes left on the clock, Luke Arscott again came agonisingly close to touching down for the visitors.
Chasing a kick through, the Bristol full-back knocked the ball on as he attempted to touch down over the Chiefs line.
Not to be outdone, Exeter fought back and went within inches of scoring themselves.
A superb line break from Phil Dollman saw the centre just stopped short as he reached for the line.
As play broke down, Exeter's pack piled forward and the home side thought they had their reward when they drove over the line, only for the referee to deem the ball was held up.
But the Chiefs were rewarded for their efforts with that late penalty, which Steenson's ever reliable boot put between the uprights to give his side that narrow first-leg advantage.
Exeter Chiefs head coach Rob Baxter told BBC Spotlight: "It was a fantastic game of rugby, I know some people don't like to see games without tries, but that had everything.
"There was a great deal of skill and a massive amount of physicality out there, and for me that makes a fantastic spectacle.
"We're going to enjoy this week, and we're going to look forward to going up there.
"After tonight's result it's pretty much going to come down to the one-off game and for us that's a fantastic position to be in."
Bristol coach Paul Hull told BBC Radio Bristol:
"We're initially disappointed with the fact we lost the game because rugby's about winning, but it's 3-0 at half time and that's not too bad to go home with.
"It was a night for defence really. I thought Exeter defended really well and we did as well. There was a lot of passion and intensity in the game.
"It was always going to be a tight game and 3-0 down at half-time, I'm pretty pleased with that."
Bristol full-back Luke Arscott told BBC Radio Bristol: "Exeter are a very physical side, they fronted up in all areas and deserve massive credit for that.
"We're not too disappointed but obviously we would have liked to take a small advantage back home.
"There's 80 minutes to go and it's effectively 3-0. We're relatively happy with that."
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