A rare try from Nathan Williams along with eight points from fly-half Jason Tovey helped give the visitors a 13-3 advantage at the interval.
But that was soon turned on its head as Griffin crossed on 56 minutes followed by Duffy ten minutes later to lift Connacht up to the giddy heights of second place in the RaboDirect PRO12 table.
It also extends the Dragons' miserable run against Connacht, who have won the teams' last seven encounters, while their last away win in the league came in February.
The omens were not good for the Dragons at the Sportsground, where they had not won for seven years, and within two minutes it looked like it might not be their day when Connacht were penalised for not releasing and Tovey's kick rebounded to safety off the post.
But Tovey did open his account after seven minutes when Connacht hands were spotted in the ruck by referee Leighton Hodges.
Niall O'Connor levelled things up three minutes later with a monster kick and then opted to kick to the corner as the hosts applied the squeeze.
That decision was nearly vindicated when winger Brian Tuohy was held up just short and there was more frustration when another O'Connor penalty drifted just wide.
But it was the visitors who seized the initiative on the half-hour mark when Matthew Pewtner's superb line took the Dragons deep into Connacht territory before prop Williams barged his way over from close range.
Tovey converted to give the visitors a 10-3 lead which he extended with a penalty three minutes later after a storming run from Martyn Thomas.
Connacht were granted a break however when Tovey's radar was slightly off with another penalty just before the break to keep the deficit at ten points.
Eric Elwood sent his troops out fired up for the second half with half-backs Matthew Jarvis and Paul O'Donohoe entering the fray and their their bright start was rewarded on 53 minutes.
Winger Tiernan O'Halloran sliced his way through the Dragons defence and his superb offload was gathered by Griffin who touched down in the corner.
Jarvis added the conversion and suddenly all the momentum was with the hosts.
That pressure eventually told when Duffy made the initial break and after a couple more phases was able to ground the ball. Jarvis' impeccable conversion gave the home team a four-point lead.
There seemed little prospect of a comeback by the away side in the closing stages as Connacht spent most of the remaining time in the Dragons half searching for a score to kill the game off.
Although that try did not arrive, a third RaboDirect PRO12 in four games represents an excellent start to the season by Eric Elwood's men.
|Connacht Rugby Score Card|
|Newport Gwent Dragons Score Card|