The victory should be enough for the Chiefs to concentrate on pushing on for a top-six spot, but Newcastle probably should start planning for life in the Championship.
They remain well adrift at the foot of the table but did squeeze a losing bonus point out of the game with Tim Swinson's converted try two minutes from time.
The Chiefs led 10-0 at half-time after an opportunist try from scrum-half Haydn Thomas as Newcastle's habit of starting slowly and gifting tries struck again.
Fly-half Ignacio Mieres kicked the conversion and added penalties either side of half-time plus a drop goal four minutes from the end to see the Chiefs cruising at 16-3 until Swinson's late strike raised a few doubts.
Gopperth converted the close-range try to add to his second-half penalty to clinch a bonus point for the Falcons at 16-10, but it came too late.
There was simply no-one at home when Thomas scampered through a yawning gap on the 22 and the scrum-half was never going to be caught, and Mieres' conversion made it 7-0 after 10 minutes.
The Falcons did have a couple of chances, the best midway through the half when Jeremy Manning just failed to get on the end of a Gopperth kick through, but they were on the back foot for the rest of the first half as their line-out misfired again and they failed to look after the ball when they had it.
Gopperth missed a long-range penalty and while they had the lion's share of possession, Exeter failed to make it count against some top-class Newcastle tackling.
The Falcons coughed up the ball once again in their own half and conceded a penalty which Mieres slotted right on half-time for the Chiefs to lead 10-0 at the break.
Gopperth kicked a penalty within a minute of the restart but Mieres was on target as well two minutes later.
And when Gopperth missed two difficult penalties in three minutes and the Falcons made a mess of three attacking line-outs in the Chiefs 22, it was hard to see Newcastle getting anything out of the game at all.
There was a brief flurry sparked by Suka Hufanga's weaving run, but once again Newcastle's inability to protect the ball ended any hopes of a score.
With six minutes left Mieres slotted his drop goal after some patient phases from the Exeter forwards.
Newcastle finally found some edge to their play and Swinson barrelled through followed by Gopperth's quickly taken conversion but it was not nearly enough.
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