Exeter fly-half Ignacio Mieres kicked the Chiefs to the top of Pool Four in the Challenge Cup with six penalties and ended the Dragons' unbeaten run.
The Argentine converted four as his side enjoyed more territory and possession in the first half.
Dragons changed tactics after the break and Stefan Jones put them within six points of the hosts on the hour.
Mieres added his fifth kick to deny Dragons a losing bonus point and landed his last at the final whistle.
Exeter enjoyed a dominant first half and exerted greater pressure on the Dragons, especially in the scrum - despite the return of Wales internationals Dan Lydiate and Toby Faletau to the Dragons' ranks.
Amid wet conditions the visitors tried to play their way out of their own half but were thwarted by the Chiefs' superior intensity.
Things were made more difficult for the Pro 12 outfit when scrum-half Joe Bedford was sin-binned on 33 minutes for an unclear offence.
The visitors changed tactics in the second period, keeping the ball tight, and started to edge their way back into the game.
Wayne Evans came on for Bedford and the visitors started to dominate and win decisions.
Jones landed his two shots at goal to put them in contention to at least take a point away from Sandy Park but Exeter copied their opponents' tactics on the hour mark and started to dominate possession.
Mieres's fifth penalty gave Exeter breathing space and while neither side looked like scoring a try the hosts were comfortable and rounded off the win at the death.
Exeter: Arscott; Camacho, Sestaret, Dollman, Jess; Mieres, Thomas; Tui, Alcott, Mitchell; Hayes, Bentley; Baxter, White, Phillips.
Replacements: Tatupu for Camacho (66), Phibbs for Thomas (69), Budgen for Tui (66), Andress for Mitchell (59), Muldowney for Bentley (59), Ewers for White (69).
Newport-Gwent Dragons: M Thomas; Harries, Riley, Tuilagi, Hughes; S Jones, Bedford; Price, Burns, Buck; A Jones, Sidoli; Lydiate, Thomas, Faletau.
Replacements: Brew for Harries (61), Tovey for S. Jones (69), W. Evans for Bedford (44), N. Williams for Price (65), Buckley for Burns (72), Jenkins for Buck (65), Morgan for Sidoli (65).
Sin Bin: Bedford (34).
Ref: Carlo Damasco (Italy).