On his Connacht debut new flanker Nepia Fox-Matamua touched down for a first half try while flying wingers Danie Poolman and Fionn Carr crossed after the break. Outhalf Jack Carty kicked 14 points from the tee.
Carty slotted the first points of the season after five minutes and a penalty won by the home side after Dragons came in at the side of a driving lineout.
When the Welsh region were accidentally offside a minute later the Athlone man doubled the tally to make it a 6-0 lead.
Dragons fullback Carl Meyer stepped up to the tee on 14 minutes but missed his attempt and Carty slotted his third shortly after when the visitors were again offside.
Connacht had an attacking scrum on 25 minutes and the dominant front-row won a penalty for Carty to kick to the corner. From there the pack, helped by some backs, barged their way over with a perfectly executed driving maul, debutant Fox-Matamua the try scorer.
Carty missed with the extras and outhalf Jason Tovey, who took over the kicking duties from Meyer, then got the visitors on the scoreboard with a penalty.
Half time: Connacht 14 – 3 Dragons
The Dragons came out charging after the break, and within two minutes had closed the gap with a try from debutant South African scrum-half Seral Pretorius. Tovey picked a line to make the initial burst and after a number of pick and drives, the number nine burrow over, Tovey's conversion making it 14-10.
Pat Lam’s men hit back almost immediately when an attacking scrum saw the ball come clear and replacement scrum half John Cooney fed man of the match Tiernan O'Halloran who put Poolman over in the right corner. Carty's conversion was off target but the lead was back out to nine points.
The Dragons were not going away though, a Tovey penalty reduced arrears to 19-13 and Meyer was just wide with an effort from inside his own half.
But as the half wore on Connacht were the dominant force, searching for winning score and after opting to kick for the corner over the sticks to no avail, Carty took on a subsequent opportunity to make it 22-13 as the game entered the last ten minutes.
A Tyler Morgan break got the Dragons back on the front foot before replacement lock James Thomas went over, Tovey's successful conversion making it 22-20 with five minutes still to play.
But the men in green grabbed the game's final try when centre Bundee Aki burst clear for an impressive break before putting replacement Carr over in the corner. Carty tapped over the conversion.
The Dragons refused to go home empty handed and grabbed a penalty from the scrum which Tom Prydie slotted as the last kick of the game to give his side the losing bonus point.
Full time: Connacht 29 – 23 Dragons