A stunning start from Exeter saw them take the lead after four through Don Armand and the boot of Gareth Steenson extend the lead to 13-0 just before the half hour mark.
Back came Blues with a brilliant converted score from Wales wing Alex Cuthbert seeing them cut the lead to 13-10 at the break.
A tense and enthralling second half saw Blues draw level, but two late penalties from Steenson ensured Blues had to settle for a losing bonus point and wait to see if other results will fall for them over the weekend to claim one of the three Amlin spots.
The game started with both teams trading kicks for territory under dark Cardiff skies with Fetu'u Vainikolo the first to break the line only to be tap tackled by Dafydd Hewitt.
Exeter went to work immediately through the forwards and slowly bulldozed up to the try line before crashing over from close range. The Chiefs celebrated the score and, after consulting the TMO, the try was given to Armand as the visitors made a dream start.
A free kick at scrum time gave Blues an attacking platform but Exeter held firm and then won a penalty of their own for offside that allowed them to kick deep into home territory.
Blues looked to launch an attack from deep but they were met by a wall of defiance from Exeter who looked to shunt them back before winning a penalty for holding on that Steenson drilled over for a 10-0 lead.
An offside decision went against the chasing Exeter backs, after a high up and under from Steenson, and Gareth Davies popped the ball in the visitors 22 for the first time in the game on 15 minutes.
Exeter though repelled any attack before turning to their power up front as they won a scrum penalty and then a driving lineout slowly battered into the Blues' 22. Only Chris Czekaj's hands stopped a floated pass from finding an extra Exeter man out wide for a second try.
Only stubborn defence stopped Exeter from extending the lead as they crashed into collisions five-metres out and rumbled up close before conceding a try as they held on.
The rain continued to lash down on the artificial pitch as Exeter pushed the lead out to 13 points with another three-pointer from the reliable boot of Steenson.
A long kick from Henry Slade caught Blues out as Halfpenny had to race back close to his line to pick up. The full back then lost control to hand Exeter a scrum five-metres out that Kai Hortsmann looked to take advantage of but knocked on as he went for the line.
Exeter put Blues under pressure at the resulting scrum and Lloyd Williams was forced to concede another five-metre scrum as he collected the ball after it popped out the back. Again Horstmann looked to the line but against Exeter were this time pinged on the line.
Blues looked to add some pace to the game and broke from deep through Halfpenny and Czekaj. A kick ahead by the wing saw Matt Jess tagged and he gave away a penalty that allowed Halfpenny to open the home account.
Great turnover work in the tackle from Macauley Cook gave the home side more ball that Gavin Evans kicked in behind but Alex Cuthbert failed to make a similar connection as he looked to hack ahead on the byline.
The home side were starting to play with invention and more urgency as a cross kick from Davies found Cuthbert before quick hands kept the move alive before Blues knocked on in slippery conditions.
The home side struck with a simple devastating move just before the break as Dicomidis spotted a gap and fed Ellis Jenkins. The young flanker was alert enough to then pop the pass inside to Cuthbert who burst clear to dive over and cut the lead to 13-10.
The Arms Park was alive as Dafydd Hewitt now broke from deep before Robin Copeland burst past a poor tackle and rattled the ball up to halfway as the teams went into the break separated by three points.
Dicomidis was in the thick of the action at the start of the second half as he pushed Exeter players back in the tackle and looked to give Blues momentum in a now tense and closely fought affair.
A game of aerial ping pong between the teams resulted in an advantage to Chiefs as Blues kicked out on the full in increasingly difficult playing conditions. But the hosts held firm as mistakes started to creep into the Exeter game.
Czekaj burst down the wing as Blues continued to look threatening with the ball in hand while a dragged down driving lineout gave Halfpenny a shot at goal from out wide that he made no mistake with to draw the teams level.
Back came Exeter and a big scrum, after a knock on the left wing, gave Steenson the chance to put the visitors back ahead and he, as Halfpenny did minutes before, made no mistake from a central position.
A charged down kick on Davies fell kindly for Blues as they swarmed the isolated Exeter man who gave away a penalty for holding. That gave Halfpenny a long range penalty attempt, but his kick fell left of the target.
A flying break and kick ahead by replacement Dan Fish sparked another counter from Blues and when the hosts rattled into contact they won the scrum put in five metres out to huge cheers. But after Copeland broke off the back of the scrum the ball was lost forward.
Blues refused to lie down and when they battled up to the line a penalty was won and Dave Lewis was yellow carded. Gareth Davies stepped up but missed the kick to the astonishment of the home crowd.
Exeter made sure of the win in the closing seconds with a penalty from Steenson after yet more impressive set piece work from Chiefs.
04m Armand t Steenson c 0-7
12m Steenson p 0-10
24m Steenson p 0-13
30m Halfpenny p 3-13
37m Cuthbert t Halfpenny c 10-13
54m Halfpenny p 13-13
59m Steenson p 13-16
77m Steenson p 13-19
Cardiff Blues: 15 Leigh Halfpenny, 14 Alex Cuthbert, 13 Gavin Evans, 12 Dafydd Hewitt, 11 Chris Czekaj (Fish 59), 10 Gareth Davies, 9 Lloyd Williams (Jones 63); 1 Sam Hobbs, 2 Kristian Dacey, 3 Benoit Bourrust, 4 Chris Dicomidis, 5 Filo Paulo, 6 Macauley Cook, 7 Ellis Jenkins (Watts-Jones 64), 8 Robin Copeland.
Replacements: 16 Marc Breeze, 17 Thomas Davies, 18 Scott Andrews, 19 James Down, 20 Rory Watts-Jones, 21 Lewis Jones, 22 Dan Fish, 23 Richard Smith