Dragons skipper Forster crashed over in the 75th minute to finally see-off the Irish province and bag a bonus point at the same time. But the Dragons will know they should never have left it so late after out-scoring Connacht four tries to one in the first half after scores from Gareth Chapman, Michael Owen and a penalty try.
The visitors opened the scoring through the boot of David Slemen on seven minutes but the Dragons' response was clinical as the backs scorched through to send wing Chapman over for his second try of the week. Chapman scored his team's third try in the New Year's Day victory over the Ospreys and finished in style after Craig Warlow's long pass to Hal Luscombe set the Dragons backs alight.
The Dragons forwards did the damage for their second try on 21 minutes. Owen was gifted the easiest of touch downs after the Dragons scrum shoved Connacht over their own try line. Yet every time the Dragons appeared to take charge they threw it away with basic errors that allowed their visitors to hit back.
David Slemen had kicked two penalties either side of the first try and then capitalised on Sione Tupulotu's missed catch to canter over just moments after the Dragons' second for a 13-10 lead. The Dragons' third came just before half-time. This time referee Ian Heard blew for a penalty try, after the Connacht scrum collapsed for a third time under pressure in front of their own posts, to give the Dragons a 17-16 lead.
Warlow and replacement Paul Warwick traded a penalty apiece as the second half descended into a closely fought affair with scoring chances few and far between. However the game finally turned in the Dragons favour when Dan McFarland was yellow carded for collapsing the scrum. Connacht held out but conceded penalty just a minute before he was due back on the field.
Forster opted for the lineout drive instead of the kick at goal and was rewarded with the crucial try.