The Dragons were leading the intensely fought encounter at the interval following Steffan Jones’ late touchdown.
But the Blues survived a scare early in the second-half and surged in front thanks to Cuthbert’s second try in as many games.
Darren Edwards’ men had their own chances but they failed to convert and Patchell put the visitors out of reach.
It was a dream first start for the 19-year-old, who starred for Wales Sevens last year, in the number 10 jersey.
Patchell said: “I had a lot of old heads around me and they helped me and gave me a lot of confidence after I missed that first kick.
“Ceri Sweeney was great all week, telling me what to expect and perhaps some of that experience rubbed off.
“Rodney Parade is a pretty intimidating place to play with a loud and aggressive crowd but I thoroughly enjoyed it. We had to dig deep but thankfully we came through in the end.
“I’ve done a lot of work over the summer on my kicking and, having a background in Sevens, taking the ball to the line is something I enjoy. It was great to get the win.
“Daf Hewitt and Gavin Evans have a lot of experience and are good players. For a young fly-half that is what you want outside you – players you can turn to.
“Jamie (Roberts) also helped me no end when he came on. He was constantly talking to me and telling me where there was space to kick.
“It’s great to see him back on the field after working so hard to get back following his injury.”
Both Patchell and Wales Under-20 star Tom Prydie missed shots at goal in a frenetic opening period.
But the young fly-half appeared to have created the first try when he ghosted through the Dragons defence. He linked with Hewitt but the centre’s pass to send Evans over was adjudged to be forward.
Jones was also unfortunate not to give the Dragons the lead having hit both the right post and crossbar with a long-range penalty.
Patchell booted the visitors into the lead after a scintillating break from Alex Cuthbert but the Dragons quickly responded.
Lewis Evans released Phil Price with a fine off-load and although he was tackled short the ball was recycled and Jones darted over.
The try handed the Dragons a 5-3 lead at the interval and they could have extended it further quickly following the restart.
Price twice came agonisingly close but the Blues were under the cosh when Nathan Trevett was shown yellow.
But despite being reduced to a seven-man pack the Blues won the crucial penalty at the ensuing scrum and momentum swung in their favour.
Wales centre Jamie Roberts made his comeback on 46 minutes and moments later the Blues regained the lead.
Adam Jones fumbled the ball on the 10 metre line and Evans capitalised. He should great awareness and slight on hand to offload to Cuthbert who raced home.
Patchell converted and added two late penalties to fire the capital city region out of reach and secure derby bragging rights.
