The Blues won the coin toss and opted to play into a strong wind in the first-half and the decision paid off as they went into the break with a 7-3 lead thanks to the centre.
Wilson’s men had further opportunities in the second-half but Connacht, fired up by their partisan crowd, fought for everything.
They regained the lead following a try from full-back Darragh Leader and despite a penalty from Steve Shingler, Shane Delahunt appeared snatched the spoils with a driving lineout with three minutes to go.
But the Blues refused to be denied and the hot-stepping centre crossed at the death, before Evans stepped up to slot the vital conversion from out wide.
Cardiff Blues now face Munster in their next encounter on the road before hosting the Dragons when they return to the Arms Park on October 6.
Despite starting the encounter with assurance, it was Connacht that had the first scoring opportunity after Willis Halaholo was penalised when on the attack for side entry.
Connacht, with the help of the wind, drilled the ball into the corner and launched a driving lineout but Cardiff Blues defended imperiously and were soon able to clear their lines.
There was not much respite however, with debutant fly-half Andrew Deegan looking lively and the Blues unable to work their way out of their own half.
They enjoyed sustained periods of pressure but were unable to break the Blues down, eventually settling for a penalty which Craig Ronaldson slotted.
The Blues displayed flashes of their own attacking prowess with Tom James, Halaholo and Rey Lee-Lo combining and Lloyd Williams sniping around the fringes but they were penalised at the breakdown once more by Marias Mitrea.
They were on the front foot once again mid-way through the half after Seb Davies latched onto loose ball and beat two men before cantering into the Connacht half.
However, he was unable to find support and the opportunity was lost as Connacht knocked the ball forward as Shingler looked to find Cuthbert with a short ball.
Minutes later Cuthbert was back in action as Connacht failed to deal with a Lloyd Williams grubber. The powerful wing dragged two defenders down the left with him but he was unable to break through.
Possession was maintained and a try appeared on the cards as Halaholo found Lee-Lo with a long pass but the Samoa centre was well defended by Bundee Aki.
The Blues remained in ascent but were soon penalised at the scrum, allowing the home side to clear.
They strung a number of off-loads together as they surged forward but Navidi was able to intercept and release Rhun Williams on the counter-attack.
The home side were dealt a blow on 32 minutes as captain John Muldoon was sent to the sin-bin but they could have no complaints.
Shingler scythed through the Irish defence and found Navidi in support. He rampaged forward and looked to release Nick Williams with a neat offload but the number eight was tackled without the ball.
Shingler drilled the penalty into the Connacht 22, but they were immediately penalised for blocking and again left empty-handed.
But there was to be no mistake on 37 minutes as they launched another attack from a penalty. Shingler popped a flat pass to Halaholo and the lively centre sliced through the defence and shimmied past Darragh Leader for a vital try.
Shingler converted the effort from in front of the sticks to give Cardiff Blues a 7-3 lead with half-time quickly approaching.
The Blues had the opportunity to stretch their lead early in the second-half after Nick Williams earned a superb penalty at the breakdown.
Shingler fired the ball deep into the Connacht 22 but Quin Roux snatched the lineout and while further opportunities were forthcoming, the Blues were unable to breakthrough as the home side defended and competed at the breakdown ferociously.
The Galway-based province rallied as the hour-mark approached and they went through more than 20 phases after the Blues conceded a penalty.
Wales Capital Region defended heroically but the pressure eventually told as they ran out of numbers and Leader crossed in the corner.
Deegan was unable to convert but the try gave the home side an 8-7 lead, however it did not last long as Shingler quickly slotted a difficult penalty.
The home side fought tooth and nail and they had one effort ruled out by the TMO, after defender Cuthbert was obstructed.
However, Kieran Keane's side looked to have secured victory when Delahunt touched down from the back of a rampant driving line-out with only four minutes remaining.
However, the Blues regained possession from the restart, and following a solid scrum the ball was eventually shipped wide to Halaholo who dummied his way past the defence to score and bring the scores level.
The pressure of the conversion was on Jarrod Evans' shoulders, but, despite the tough conditions, the replacement outside half kicked the ball into the win to slot the match winning extras.