The Blues moved above Edinburgh and Treviso into seventh place thanks to claiming just their third win in their last 14 matches in all competitions.
They were under the cosh for much of the match, but pulled through thanks to four penalties from teenage fly-half Rhys Patchell.
The Dragons claimed the only try of the contest through centre Andy Tuilagi but were left to rue other missed chances during a game played in appalling conditions.
Despite playing into the wind and driving rain, it was the visitors who were to have the better of the first half in terms of possession and territory, with their set-pieces on top.
But they were unable to convert that pressure into points, with outside-half Dan Evans sending an early penalty attempt to the right of the posts and also failing with a later effort.
In contrast, the Blues were to take their chances when they came along.
From virtually their first entry into opposition territory on 15 minutes, they took the lead through Patchell, who struck a fine angled 45 metre penalty after Dragons winger Mike Poole pushed Dafydd Hewitt into touch off the ball.
The 19-year-old Patchell was to strike twice more as the conditions worsened still further, with the rain turning torrential midway through the first-half.
On 21 minutes, he doubled the lead after Wales skipper Sam Warburton won a penalty with some excellent breakdown work, as Dragons lock Andrew Coombs was done for holding on under pressure from the flanker.
Then, just before the half hour mark, Patchell landed his third successful shot at the sticks after the Dragons went offside after an attempted clearing kick was caught up in the fiercely gusting wind.
His opposite number Evans had an immediate opportunity to hit back when the Dragons were awarded a penalty right in front of the posts courtesy of their dominant scrum.
But in attempting to drive his kick through the wind, Evans sent it wide to leave his team trailing 9-0 at the break, despite having had the upper hand in most respects.
The question was would a nine-point lead be enough, with the Dragons having the elements in their favour in the second half?
It was not long before that advantage was significantly reduced as Tuilagi scored the opening try of the game for the visitors on 48 minutes.
After sustained pressure on the Blues line, Tuilagi took a short pass from Poole and smashed through winger Harry Robinson, taking two players to the line with him.
Scrum-half Wayne Evans added the conversion with Dan Evans injured and it was the latter's replacement at fly-half, Lewis Robling, who put the Dragons ahead for the first time on the hour mark.
Their lead was short-lived however as Patchell promptly landed his fourth penalty, making light of the wind in his face.
And, despite piling on the pressure, the Dragons were unable to conjure up the winning score, with Robling just wide with a late drop goal attempt.
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