The Chiefs found themselves 16-6 down midway through the Amlin Challenge Cup clash at Rodney Parade following a stunning try from Toby Faletau.
However super-sub Gareth Steenson came off the bench to help produce an impressive fightback against their Welsh opponents.
Steenson kicked a penalty and also converted tries from Simon Alcott and James Phillips, who crossed in the 77th minute for the game-winner.
Jason Tovey, back in the Dragons' starting line-up, booted the hosts into the lead after just six minutes and exchanged further penalties with Exeter fly-half Ignacio Mieres as the Welsh region edged into a 9-6 lead after 24 minutes.
But with Exeter's Alcott sent to the sin-bin, the Dragons surged into a commanding lead thanks to a great score from Faletau.
The Wales international burst through the visitors' defence from his own 22 and after combining with Tovey and Aled Brew, crashed over the line.
Tovey, who finished the night with a personal haul of 14 points, landed the extras to give the hosts a sizeable cushion at the break.
Exeter, though, came storming back, starting with hooker Alcott making amends for his first-half yellow card by going over on 51 minutes.
Steenson continued to chip away with penalties and Chiefs then snatched the victory at the death when Phillips dived over following a dominant final quarter.
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