Nathan Williams' first try for the Welsh region was the decisive score in the first half of the RaboDirect PRO12 clash as Darren Edwards' men went in at the break leading 13-3.
Eric Elwood had demanded better discipline from his Connacht side after their loss at Edinburgh last weekend, but they gave some soft penalties to the Dragons.
Jason Tovey kicked two of his four penalty attempts, while Niall O'Connor's right boot was responsible for Connacht's only points until Griffin touched down coming up to the hour.
The converted try lifted Elwood's men and with Matthew Jarvis adding the extras to Duffy's touchdown, the four-point margin was enough.
Tovey watched his first penalty attempt bounce back off a post, but from a similar position he gave the Dragons a seventh-minute lead.
In wet conditions, O'Connor levelled for Connacht with a well-struck penalty from distance and Ray Ofisa and Henry Fa'afili began to wield some influence in the loose.
O'Connor was off target from a central penalty, and the necessary penetration was lacking as both sides struggled to build momentum in attack.
With the wind proving a factor, O'Connor turned down a kickable penalty in the 23rd minute and some good approach work was foiled by an accidental offside.
The Dragons were patient and struck impressively for a 30th minute try. Matthew Pewtner popped up in the middle on an angled run that took him past Ethienne Reynecke, and the pressure eventually told as prop Williams crashed over.
Tovey converted and added a penalty after his initial break, with Connacht suddenly 13-3 adrift - it would have been worse had the young out-half not miscued a late shot at the posts.
The second half stuttered along, with Brian Tuohy the most threatening of the home backs as they tried to hit back at the Welshmen.
Suddenly, replacement half-backs Paul O'Donohoe and Jarvis set the Connacht attack in motion, terrific hands down the short side seeing Griffin link with Tiernan O'Halloran and gather the return pass to score in the corner.
Jarvis followed up with a brilliant touchline conversion to close the gap down to three points, and with the 3,706-strong crowd raising the decibel level Connacht snaffled a second try.
Duffy sniped through with ball in hand and after good continuity from the forwards, O'Halloran's inviting pass put his captain diving over in the corner.
Jarvis again had the accuracy from the left touchline. The 21-year-old missed a subsequent penalty but the Dragons' challenge had petered out.
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