Steffan Jones' first-half try was the only score of the game for the home side while in addition to Cuthbert's try the impressive Rhys Patchell added a conversion and three penalties.
The other bright spot for the Blues was the return in the second half of Jamie Roberts after six months on the sidelines with a knee injury.
The opening stages were a story of misfiring boots as Patchell for the Blues and Jones and Tom Prydie for the Dragons all missed makeable penalties.
But the game came to life in the 25th minute when Blues full-back Dan Evans looked in for the first try of the game but he was called back after the impressive Patchell's offload was adjudged to have gone marginally forward.
Still the scores were locked at 0-0 as four minutes later Jones' penalty attempt came back off the posts.
Both sides were having their chances and a fantastic counterattack from the Blues was only ended when Evans' chip over the top was slightly overhit.
But when the first score did eventually come it was Patchell who made up for his earlier miss by slotting home a penalty.
Having waited virtually the whole half for the first points the first try arrived almost immediately after as the Dragons broke through, fly-half Jones ghosting over to the right of the posts after stepping inside his man.
Shortly after the interval the Dragons looked like they had extended their lead with a pushover try but the Television Match Official ruled the replays inconclusive, though Blues prop Nathan Trevett was sent to the sin bin.
The Dragons were desperate to go for a pushover score from the subsequent five-metre scrum but the Blues held out and it was the home side who conceded the penalty in the end and the chance had passed.
And in the 48th minute despite a man advantage they were made to pay when Adam Jones' knock-on was collected by Evans and his quick offload set Wales international Alex Cuthbert in the clear to cross in the right-hand corner, Patchell adding the extras to make it 10-5.
The Cuthbert try also precipitated the introduction of Roberts for his first appearance since April after knee surgery.
But back came the Dragons and Prydie had the chance to reduce their arrears in the 62nd minute but his penalty from the right touchline drifted narrowly wide.
Chances were very much at a premium however as the game wore on and when Patchell fired over two penalties in the 76th and 78th minutes the match was lost for the Dragons.
|Newport Gwent Dragons Score Card|
|Cardiff Blues Score Card|