Despite taking an early lead, the Blues trailed 14-6 at the interval but they displayed an abundance of grit and determination as they battled back.
Willis Halaholo skipped through the defence to claim the crucial try – his fourth since joining the region – and Shingler kept the scoreboard ticking before slotting the match-winning kick to take his haul to 17-points.
The victory keeps the Blues' hopes of reaching the European Challenge Cup quarter-finals alive but following Bath's bonus-point victory over Bristol, top spot is out of their hands.
The Blues made an assured start in France and Shingler booted them into the lead with a simple penalty after just three minutes.
But it did not last long as Pau responded with the opening try just a minute later. The damage was done when the hosts burst through midfield, and after retaining possession they took advantage of an overlap on the right to send Louis Dupichot over in the corner.
Brandon Fajardo slotted the conversion from wide to make it 7-3 after six minutes before Shingler added a second penalty to cut the deficit to a single point.
The Blues were enjoying plenty of territory in the encounter but opportunities were limited and Pau soon claimed a second try.
With an abundance of powerful ball carriers they rampaged out of their half to earn a penalty. Fajardo dinked it inside the Blues’ twenty-two and his pack set up a rolling maul.
It was initially suppressed but eventually picked up a head of steam before a number of pick and drives created the opening for scrum-half Julien Tomas to dart over. Fajardo added the extras to open up a 14-6 lead.
Mistakes continued to hamper the Blues and they were dealt a blow when Nicky Robinson was forced off with 30 seconds of the half remaining.
Shingler booted the scoreboard back into action just three minutes into the second-half to make it 14-9.
Like in the first-half, the Blues were enjoying plenty of territory but on this occasion they made it count.
A dazzling break from Morgan made major inroads and he found Macauley Cook in support. Thinking Cory Allen was on his outside he passed right with no one there but the possession was maintained .
Pau finally intervened but some loose passing and huge pressure from Garyn Smith forced them over their own line.
Despite huge pressure at the scrum, Josh Navidi carried and presented the ball and with a switch of direction Shingler found Halaholo, who eased through the French defence to score.
Shingler added the simple conversion to take the lead for the first time since four minutes into the game.
Wales’ Capital Region continued to take hold of the game with Sam Warburton carrying to good effect and linking with Macauley Cook.
Another moment of brilliance from Morgan unlocked the Pau defence once more as he eased around the defence before offloading around the corner but Rhun Williams was unable to hold onto the audacious pass.
However they came away with points a minute later as Shingler maintained his 100 per cent strike-rate with a fourth penalty.
Pau responded with venom and turned the screw with a sustained period of pressure in the final corner. With a strong set-piece they kicked two penalties to the corner and while the Blues held firm initially, it took its toll and Pierre Dupouy crashed over.
Fajardo converted to reclaim the lead leaving Cardiff Blues with little more than 10 minutes to snatch the result.
They battled tooth and nail, with Shingler, Morgan and Allen growing in prominence and threw everything at Pau. And with two minutes remaining they were awarded a penalty and Shingler displayed nerves of steel to snatch the win.