For Connacht, their poor form on the road continued and despite a bright start, Eric Elwood's men saw their high penalty count cost them dearly, ensuring they have now lost three in a row on their travels in the RaboDirect PRO12, and failed to reach double figures in each.
The visitors began the brighter and made their early possession count, capitalising on some Dragons indiscipline as Dan Parks knocked over his one and only penalty of the night on six minutes.
At a rain-soaked Rodney Parade, free-flowing rugby was at a premium and it took until 25 minutes into the match for the next scoring action - RaboDirect PRO12 top-scorer Tom Prydie notching a penalty of his own after Connacht hooker Jason Harris-Wright was sent to the sin bin.
And Connacht were in all sorts of trouble when Andrew Browne joined him on the half-four mark - and the Dragons, shorn of confidence in recent weeks, made them pay.
On 32 minutes, Prydie missed his second penalty kick of the night but just minutes later the Dragons capitalised on their numerical advantage to send centre Leach over the try-line.
Prydie missed the conversion however, meaning Connacht, who introduced Ethienne Reynecke for Willie Faloon, were still in touching distance at half-time, 8-3 down.
Buoyed by taking the lead into the break however, the Dragons swiftly extended their advantage in the second half with Prydie rediscovering his accuracy with the boot to add another penalty.
And the Dragons were able to open up some considerable daylight on 52 minutes when Prydie again showed his ability from the tee - stretching the score to 14-3 in the hosts' favour from straight in front of the posts.
The Dragons could scent victory and they set about adding to their lead and after Parks wasted the chance to reduce the arrears with a penalty, Dan Evans also unleashed a fine drop goal attempt that missed by just inches.
Lewis Evans was sent to the sin bin with 13 minutes remaining, ensuring some nervous few moments for the Dragons faithful but Connacht could not find another way through, bringing some much-needed cheer to Rodney Parade.
|Newport Gwent Dragons Score Card|
|Connacht Rugby Score Card|