In sun-kissed Belfast – a huge contrast to the wind and rain that had impacted Ulster’s last-16 and quarter-final wins – the opening stages of the game brought plenty of intensity but was littered with handling errors from both sides.
Doak, usually so reliable from the tee, missed a penalty from just outside the 22 in the 10th minute.
Ulster thought they had made the breakthrough their early pressure deserved when Michael Lowry scampered over, but it was ruled out for a neck roll from Cormac Izuchukwu earlier in the play.
Similar to Doak, Exeter missed a big chance to get on the board when Harvey Skinner sent a penalty wide, and Ulster soon got the first try of the game in the 26th minute.
With another penalty advantage, South African back row Augustus crashed over from close range and Doak added the conversion.
Ulster then lost Stockdale to the sin-bin after he caught Immanuel Feyi-Waboso high with a tackle, and Exeter immediately capitalised as Vintcent crashed over and Skinner levelled the game.
Despite having 14 players, Ulster pressed again and were held up over the line, but undeterred they came again and, after Exeter conceded their fifth penalty of the half, Doak kicked the hosts into a three-point lead at the break.
The hosts made an electric start to the second half when, after the forwards got close to the line, Doak sprayed the ball wide and Jack Murphy raced over to extend the lead, and his half-back partner nailed the conversion.
Another chance to move further ahead was lost when Augustus was held up over the line, but Ulster’s third try came from the resulting kick when Ethan McIlroy, on for Stockdale at half-time, surged into the Exeter 22 and the ball was recycled for McCann to barge over.
Despite an improved season, Richie Murphy’s side have had a tendency not to kill teams off, but Stewart’s try, and Doak’s superb conversion from out wide, put the hosts four scores clear with 14 minutes remaining.
Ben Hammersley was sent to the sin-bin with seven minutes remaining for a deliberate knock-on to rub salt into the Chiefs’ wounds, and Ulster had more than enough in the tank to get over the line despite Ridl’s consolation score.
The only concern for the hosts was a potential hamstring injury to influential centre Stuart McCloskey, who pulled up with 10 minutes remaining after kicking the ball, and Murphy will be hoping he will be fit for the final later this month.