The defending champions held off a late fightback from the Welsh side, whose defeat leaves their own campaign hanging by a thread.
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The hosts started the match strongly and went over for the first score after just seven minutes as Adam Hughes latched onto a loose Dragons pass before racing almost half the length of the field to touch down.
Hughes and Hadyn Thomas then combined to set up to Fetu'u Vainikolo, and the Tongan winger charged over to score in the corner. Ceri Sweeney was unsuccessful with his kick, however, leaving the Chiefs 12-0 ahead.
Tom Prydie reduced that deficit with a penalty shortly afterwards, but Sweeney restored the 12-point advantage with a kick of his own minutes later.
The Dragons were then reduced to 14 men as Lloyd Fairbrother was sent to the sin bin for repeatedly collapsing the scrum, and Exeter won a penalty for their troubles, which Sweeney slotted over to edge his side further ahead.
Prydie added three points for Newport before the half was over, but Exeter began the second period well and scored their third try of the match through Tom James five minutes after the restart.
The Chiefs claimed a bonus point with their fourth try shortly after the hour mark, Sam Skinner going over from close range to extend his side's lead to 30-13.
Any hopes of a comeback were extinguished with just one minute left on the clock, however, as some heavy pressure from the scrum resulted in a penalty try, while Sweeney's extras opened up a 12-point gap.
The Dragons did get one score back as Jason Tovey went over late on, but it was not enough as Exeter's title defence continued.