Two tries from Jamie Blamire, one from Philip van der Walt and two conversions and a drop goal from Ethan Grayson proved enough for Steve Diamond's side.
Exeter, without a league win in five so far this season, remain above Newcastle in the table thanks to bonus points earned.
Director of rugby Diamond said: "I'm delighted to get the monkey off the club's back from the longest run of losses, but I can't do anything about that, I've been here six months. I am trying to build something which is taking time, and hopefully this is just the start. The job in hand is putting in performances like this at home. This is what I expect.
“We out-Exetered Exeter, and a win tonight was crucial for us to stay near someone in the league. Calum Chick and Jamie Blamire’s leadership was excellent, players who if you cut them in half bleed Newcastle through and through.”
Newcastle take on eighth-placed Gloucester on Saturday and Diamond says he will wait until after that match when the first block of six Premiership matches is over before deciding whether to boost his injury-hit squad.
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