Chances were few and far between in the opening 40 minutes, but the home side managed to open up a lead, with Carty adding three penalties in the first half.
Craig Willis got the visitors on the board shortly before half time, and they closed the gap further when Richard Mayhew charged across for the only try of the match for the visitors, leaving Newcastle just two points behind.
The hosts were reduced to 14 men for a time as Rory Parata had a spell in the sin bin, but Carty added two more penalties to keep Connacht ahead.
The home side clinched the win with just over five minutes remaining as a lengthy build-up set up Danie Poolman for the try, with the winger forcing his way through the gap.
The results keep Connacht unbeaten in the competition so far this season, while Newcastle slip to their second loss.