A well worked try from hooker Kristian Dacey gave Blues a deserved early lead, but an intercept score from home skipper Jonathan Spratt on the stroke of half-time gave Ospreys a slender advantage at the break.
A tight and closely contested second half saw both teams look to strike a decisive blow before Cook stepped up with less than ten minutes to go as he pounced on an error at a home scrum to flop on the loose ball and score.
A coolly taken drop goal from Simon Humberstone gave a young Blues side – who without a total of 20 players for the clash - crucial breathing space as they wrapped up a win to reignite their cup campaign after opening pool defeat to Worcester.
The hosts went straight on the attack with Aisea Natoga looking to use his power down the left wing and fly half Matthew Morgan kicking in behind to give the Blues lineout an early examination under pressure.
Blues were rattling into the first up tackles as Richard Smith took Jamie Murphy down after a high box kick while Morgan was greeted by a thumping combination of Benoit Bourrust and Sam Hobbs as he looked to step inside.
Robin Copeland showed all his pace as he picked up a loose kick ahead that bounced kindly into the Number Eight’s hands before quick hands down the line opened up space for Dan Fish to nudge a grubber in behind Ospreys.
A penalty for offside gave Blues a first shot at goal and Simon Humberstone put the visitors ahead with a confidently struck kick between the uprights.
The impressive Dacey spun out of a tackle as the Blues forwards looked to work around the fringes, but a poor high kick ahead saw Ospreys return the ball with interest as the visitors lost valuable territory.
Morgan sliced an ambitious drop goal wide before a penalty against Blues, as they came in at the side as they looked to halt a driving maul, saw the home fly half hit the left post with a penalty attempt.
Humberstone, by comparison, made no mistake with a kick of his own to double the Blues advantage.
Gavin Evans produced a huge tackle in midfield to dislodge the ball and looked to set up an attack as he hacked ahead, only for referee Greg MacDonald to come back for a knock on by the centre.
A searing home attack down the right wing almost saw Ospreys break clear and when play was brought back it was Morgan who this time got the home side on the score board.
Home full back Murphy just about managed to deal with a skidding low kick from Fish. But from the resulting lineout the visitors menacingly drove up close before Dacey touched down.
A soft penalty against Blues, as they went over the top of the ball to protect it, saw Morgan miss another kick at goal as he dragged his effort left. But the livewire fly half made amends as his break secured another penalty that he slotted from in front of the posts.
A period of sustained Blues pressure saw them keep the ball alive and go through phase after phase for several minutes. But after slowly working up to the try line and knocking on the door, the hard work was undone with a penalty for holding on.
Ospreys then rubbed salt in the wound as they scored from nothing a minute from half-time when a loose pass was picked off by home skipper Spratt who ran in unopposed from halfway to give the hosts a lead at the break.
Errors hurt Blues in the opening minutes of the second half as they lost territory and possession in promising positions as the slippery ball proved hard to control.
Ospreys upped the tempo, as Morgan pushed a third kick of the night wide left, while the whistle of referee MacDonald was starting to hurt Blues as they looked to try and build momentum.
Fish limped off to be replaced by Tom Williams as the game entered the final half hour finely poised.
Natoga just failed to collect a neat cross field kick from Morgan that would have sent the winger away as both sides hunted what looked like being a knock-out blow in a tight encounter.
A brilliant break away from Richard Smith saw Blues race up into Ospreys territory and when they went through the hands across the field it was Williams who looked to cut back inside. Copeland then collected a pass at pace but play was taken back for going forward.
A big scrum from Blues forced a knock on deep in the home 22 and gave the visitors a big attacking platform to strike from. A quick tap and go from Copeland again saw Blues go close before referee MacDonald gave the put in to Ospreys.
Another big scrum saw Ospreys pushed back and when the ball popped out over the line it was Cook who reacted first to score to touch down.
Suddenly it was all Blues and more thrilling attacking play saw the visitors rumble up before Humberstone showed his cool to drop in the pocket and slot the drop goal.
Back came Ospreys and a series of lineouts on the Blues line saw Copeland yellow carded for an infringement at the breakdown. But just as the hosts looked set to strike with a numerical advantage a poor throw lost them possession.
Evans brilliantly ripped ball out of the tackle as Blues gave everything in the final minutes to defend the lead and secure a first win of the LV=Cup campaign.
Timeline:
08m Humberstone p 0-3
17m Humberstone p 0-6
22m Morgan p 3-6
24m Dacey t 3-11
29m Morgan p 6-11
39m Spratt t Morgan c 13-11
74m Cook t Humberstone c 13-18
77m Humberstone dg 13-21
Cardiff Blues: 15 Dan Fish (Williams 60), 14 Richard Smith, 13 Gavin Evans, 12 Dafydd Hewitt, 11 Chris Czekaj, 10 Simon Humberstone, 9 Alex Walker (Jones 60); 1 Sam Hobbs (Davies 76), 2 Kristian Dacey (Williams 65), 3 Benoit Bourrust, 4 Macauley Cook, 5 James Down (Normandale 53-57), 6 Ellis Jenkins (Hamilton 65), 7 Thomas Young (Watts-Jones 52), 8 Robin Copeland.
Replacements: 16 Rhys Williams, 17 Thomas Davies, 18 Taufa’ao Filise, 19 Miles Normandale, 20 Rory Watts-Jones, 21 Luke Hamilton, 22 Lewis Jones, 23 Tom Williams