A scintillating opening from the hosts saw them take total control of the fixture as prop Gordon Reid barged over early on before Tim Swinson scored a second on the stroke of half-time.
Danny Wilson’s men slowly repaired the damage in the second half with Dan Fish scoring a fine solo effort after collecting his own chip in behind.
Glasgow hit back immediately as replacement Simone Favaro was driven over three minutes from time, after Aled Summerhill was harshly yellow carded.
But the final word went to brave Blues in spectacular fashion as Jenkins finished off a driving maul before, in the fourth minute of injury time, Cook was at the end of a sweeping team move to dot down and claim the most unlikely of points.
It was still a defeat for the visitors to take and they will know the damage was largely done in an emphatic opening 40 minutes from the impressive Warriors.
However, once again the heart and commitment in the ranks at Cardiff Blues could not be questioned as they played for the full 80 minutes – and beyond.
The losing bonus point was undoubtedly a huge lift and - with upcoming home games against Munster and Treviso next up - hopes of climbing the Guinness PRO12 table remain strong.
Glasgow started the game in explosive fashion as lock Greg Peterson broke a tackle in the opening three minutes and was only stopped by a last ditch Patchell tackle as he motored towards the line.
The home side were rewarded for the electric start when Weir nudged over a penalty before the game’s opening try arrived on 11 minutes.
Prop Fraser Brown brushed off men in the 22 before front row partner Reid burrowed over from close range.
Glasgow were combining power and precision with devastating effect and looked to have scored a second when centre Mark Bennett strode over only to inexplicably lose control of the ball as he looked to dot it down.
As the half wore on it was slowly becoming an end to end affair as visiting scrum half Tomos Williams broke into space in the home 22 and kicked in behind before Glasgow countered from deep and Patchell was forced to drag down a marauding and side-stepping Bennett.
Weir slotted his second penalty of the afternoon to extend the lead at the midpoint of the half.
The visitors were enjoying long spells of possession and territory, but they could find no way to penetrate a rock-solid and well-drilled Warriors defence.
Summerhill and Fish combined down the left before Lee-Lo popped up down the opposite flank as Blues continued to pose questions.
Up the other end, big home tighthead prop Zander Fagerson crunched through tackles to rattle down the middle before the ball was spun wide but centre Alex Dunbar knocked on.
The visitors were putting in a big defensive shift, with Matthew Rees working hard to wrap up his man, stop the home offloading game and win a penalty just before half-time.
But there was no stopping the home side deep into injury time as wave after wave of attack finally splintered the visiting defence and Swinson blasted over with Weir’s conversion making it 20-0 at the interval.
A clearly groggy James Down – after a nasty clash of heads - was replaced at the break by Cook, with Josh Turnbull moving into the second row.
Three more changes soon followed as Josh Navidi, Taufa’ao Filise and Garyn Smith all came on as Wilson looked to fire up his troops up.
Patchell got Blues on the scoreboard after a penalty against Glasgow before a kick in behind for Fish to chase forced the Warriors to concede more ball.
Glasgow were being forced to play from deep and now work with limited possession as the visiting pack fronted up while the backs worked tirelessly to try and breakthrough.
There were reasons to be positive for the future as Corey Domachoski made his senior debut ten minutes from time before Cardiff Blues finally for the try their endeavour deserved.
A superb attack saw Blaine Scully pop up as first receiver to pull in tacklers before, when the ball was shifted down the line, Fish spotted the space in behind, chipped ahead and showed is pace to win the race to score.
The home side struck for a third time when, moments after Summerhill was yellow carded for taking a man out in the air, a driving maul shunted Favaro over.
But it was battling Blues who had the final say as Jenkins was pushed over from the lineout drive before, in a frantic finale, home lock Scott Cummings was yellow carded and Cook was the man out wide to score deep into injury time.
07m Weir p 3-0
11m Reid t Weir c 10-0
21m Weir p 13-0
40m Swinson t Weir c 20-0
56m Patchell p 20-3
73m Fish t 20-8
75m Favaro t Weir c 27-8
79m Jenkins t Patchell c 27-15
80m Cook t 27-20
Cardiff Blues: 15 Rhys Patchell, 14 Blaine Scully, 13 Aled Summerhill, 12 Rey Lee-Lo (Millard 76), 11 Dan Fish, 10 Jarrod Evans (G Smith 48), 9 Tomos Williams (Jones 61); 1 Thomas Davies (Domachowski 69), 2 Matthew Rees (E Lewis 69), 3 Salesi Ma’afu (Filise 48), 4 Jarrod Hoeata, 5 James Down (Cook 40), 6 Josh Turnbull, 7 Ellis Jenkins, 8 Manoa Vosawai (Navidi 48)
Replacements: 16 Ethan Lewis, 17 Corey Domachowski, 18 Taufa’ao Filise, 19 Macauley Cook, 20 Josh Navidi, 21 Lewis Jones, 22 Garyn Smith, 23 Harri Millard