Ulster kept their Champions Cup hopes alive by the narrowest of margins by registering a vital first win to see off Exeter, whose own European ambitions now look all but over in Pool Five.
In a tense and scrappy encounter which only saw one try, scored by Ulster's Sean Reidy in the first half, the game ended with a dramatic drop goal shootout between Paddy Jackson and Exeter's Ulsterman Gareth Steenson in the closing minutes.
Steenson first put the Chiefs two points in front with four minutes to go, only for Jackson to hit back immediately with a drop goal of his own to put Ulster 19-18 ahead.
The game then finished with Steenson going for another drop to snatch the win, but his effort went wide and the Chiefs were left with their losing bonus point.
Les Kiss' men made off with four points, though, thanks to Reidy's try, three penalties, a conversion and that late drop goal from Jackson after the home side had led 10-6 at the break, with Charles Piutau having created the touchdown with a rare moment of creativity in the game.
All the Chiefs' points came from Steenson's boot thanks to his drop goal and five earlier penalties, though the result leaves them bottom of Pool Five with Ulster in third and six points behind leaders Clermont.
Exeter kicked off after a minute's silence in memory of Anthony Foley was observed, with the supporters' own tribute then following with a rendition of the Fields of Athenry.
And it was Ulster who struck first when the Chiefs strayed offside, although Jackson's fourth-minute penalty which was equalised two minutes later by Steenson after Reidy had been caught offside.
But Ulster came again with a great attack which swung right, left and right again only for the Chiefs to be nailed for offside again just as Franco van der Merwe lost the ball on the line.
Jackson hit the corner and then opted to do so again after the Chiefs were offside at the lineout. The home side secured another penalty after Jackson's chip for Luke Marshall was called back.
Yet again, Ulster went for the corner and the lineout maul only for Rodney Ah You to knock on and the Chiefs relieved the pressure.
Exeter then spent some time in Ulster territory which yielded them two scrum penalties and a third when the home side were punished for being offside at a ruck, which saw Steenson slot a great penalty from out on the right after 28 minutes.
Then out of nothing, Piutau's hitch-kick out on the right saw him beat three defenders before Reidy followed up to drive over the line. Jackson's conversion made it 10-6 to Ulster, which is how the half finished despite Exeter ending the stronger.
Piutau helped win Ulster an early penalty after the restart but after going for the corner, the home side's maul was driven into touch.
However, from their next kickable effort, Jackson put the ball between the posts from long range to extend their lead to 13-6.
That was cut to 13-9 on 51 minutes from Steenson's third penalty of the night before Jackson missed with a 62nd-minute effort from out on the right shortly.
The Chiefs then made it a one-point game in the 66th minute after a great break from Olly Woodburn into Ulster's 22 resulted in Steenson's fourth penalty.
The scrappy nature of the game continued but when the Chiefs were pinged at a ruck Jackson gladly took the points in the 70th minute to put Ulster 16-12 in front.
But back came Exeter and when sub Rob Herring went off his feet, Steenson made closed the gap again with his 73rd-minute strike, and then he nudged the Chiefs in front with a well-taken drop goal in the 76th minute.
Stung into action, Ulster surged upfield and Jackson then put them a point in front with his own drop goal.
With two minutes on the clock, Steenson tried again but this time his drop sailed just wide and Ulster held on.