Haydn Thomas's first-half try helped Exeter overcome Premiership strugglers Newcastle at Kingston Park.
Thomas went over before Ignacio Mieres added the extras before kicking over a penalty to give Exeter a 10-0 lead.
A Jimmy Gopperth penalty soon after the break made it 10-3 before Mieres secured his second penalty and then a drop-goal to put the Chiefs in charge.
Tim Swinson scored a late try for Newcastle as Gopperth's conversion secured Falcons a losing bonus-point.
The win leaves the Chiefs fifth in the Premiership table and firmly in the running for a Heineken Cup place for next season.
They went ahead through Thomas's try after the scrum-half ran clear for Exeter's solitary try before Mieres converted and then slotted over a penalty to give the visitors a 10-0 lead at half-time.
Gopperth then kicked Falcons' first points of the match before Mieres added his second penalty.
The Newcastle fly-half missed two more attempts to kick over the posts before Argentina international Mieres' drop-goal looked to have sealed Exeter's win.
The hosts fought back two minutes from time as Swinson barrelled through before Gopperth converted but it proved to be too late for Newcastle as they were left to make do with a solitary point and now look increasingly likely to be playing Championship rugby next season.
Newcastle head coach Alan Tait:
"We have to face up to reality and we have just had a meeting with the players and coaches about where we are.
"It's going to be a tough old battle now and we have to be realistic. I have to look at myself as well and I will do that and I will have a talk with the owner.
"I'll reflect on things and on my performances over the next few days. I had a dream and vision for this club and it hasn't quite worked and I have to speak to the owner and see what happens. The knives will be out for me and quite rightly so.
"Maybe I'm writing myself off but it's hard not to do so after a result like that and things like that can cost you your job."
Exeter head coach Rob Baxter:
"We set out to be ambitious but we were very well aware of the implications for both ourselves and Newcastle.
"We knew they had said how important the game was to them and we knew we had to stop them coming out of the blocks quickly.
"We had to base things about getting a good start and hurt their ambitions and we did that. We aimed to keep the ball and to be the first on the attack and that went very well for us and Haydn Thomas scored a fantastic solo try.
"That got us on the front foot and then the game was a pretty even contest. We controlled large parts of the game without really breaking through and we weathered the storm in the second half and when we did break out, I think that was the game decided.
"It was little disappointing to kick the ball away with four minutes left and give them a try but all credit to Newcastle to attack for the losing bonus point and getting it. It was a really tough game, it's not easy to come here and get a result but I think we were deserved winners."
Newcastle: Goosen; Fielden, Hufanga, Fitzpatrick, Manning; Gopperth (capt), Pasqualin; Sheills, Vickers, Murray; Fondse, Swinson; Tu'ifua, Welch, R Mayhew.
Replacements: Bedford for Goosen (60), Catterick for Hufanga (73), Pilgrim for Pasqualin (60), M Mayhew for Vickers (58), Hogg for Fondse (41), Wilson for R Mayhew (64).Not Used: Wells, Hall.
Exeter: Arscott; Camacho, Sestaret, Shoemark, Jess; Mieres, Thomas; Sturgess, Clark, Tui; Hayes (capt), Muldowney; Johnson, Scaysbrook, Phillips.
Replacements: Barrett for Thomas (63), Alcott for Clark (73), Mitchell for Tui (41), Ewers for Phillips (67).Not Used: Andress, Bentley, Steenson, Tatupu.
Referee: Andrew Small (RFU).