The visitors had to settle for three penalties from fly-half Matthew Morgan as they were well beaten.
The importance of league competitions was again seen at Sandy Park as Exeter, fifth in the Aviva Premiership, and Ospreys, unbeaten at the top of the RaboDirect Pro12 table, fielded numerous young players for this tie.
Ospreys applied the early pressure but it was the Chiefs who got the first points on the board as skipper Gareth Steenson landed a 40-metre penalty with less than 10 minutes on the clock.
The visitors again tested the Chiefs' defence who then strayed offside allowing 19- year-old Morgan to level the scores with a kick from 25 metres straight between the posts.
Exeter began to pin the Ospreys back into their own half and when they were awarded a penalty close to the line, they opted for a scrum which collapsed when being pushed back.
Plymouth-based referee David Rose had no hesitation in awarding a penalty try which fly-half Steenson converted.
But Ospreys always looked threatening and when wing Eli Walker broke through 18-year-old full-back Jack Nowell stopped him in his tracks.
Exeter increased their lead with 10 minutes of the half remaining, after Rose consulted TMO Ed Morrison, and prop John Andress claimed the touchdown after a couple of drives to the left corner, with Steenson adding the extras.
Ospreys reduced the arrears with four minutes of the half remaining as Morgan landed his second penalty but it was the Chiefs who were showing the more attacking flair as they led 17-6 at the break.
The Chiefs wasted no time stretching their lead when Ospreys were penalised for boring in at the scrum and Steenson landed his second penalty of the night with a 40-metre kick from the left.
Errors began to creep into Exeter's game and Ospreys reduced the arrears through Morgan's third penalty from 25 metres, but this quickly cancelled out with Steenson's 35-metre penalty.
Exeter soaked up the pressure as the game drifted into the final quarter.
The home side won a turnover and were able to clear their lines and win a penalty on the Ospreys 10 metre line.
With the game virtually in the bag the Chiefs kicked for the right corner before setting up a drive to the line, but they were then penalised for not releasing the ball.
Exeter were not going to rest on their laurels as wing Mark Foster linked up with replacement full-back Myles Dorrian and replacement centre Josh Tatupu to put outside centre Phil Dollman in for their third try.
Steenson was unable to convert from out wide on the left - his first miss of the evening - but he made amends by converting the Chiefs' fourth try of the night a couple of minutes later from Tatupu.
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