Harriers ended a four-match losing streak against their bogey side with a fine battling display, which also ended Kettering's 11-match unbeaten run in all competitions.
Courtney opened his account for the Aggborough outfit to set them on their way to their first victory at the Poppies since 1993.
But it was midfielder McDermott who had the biggest impact on the visitors.
The former Walsall man had been dropped to the bench in favour of captain Chris McPhee, who was back after serving a three-match suspension.
But he changed the game when he came on for Darryl Knights at the start of the second half.
Yates said: "Darryl [Knights] was having one of those days when nothing went for him and I thought David [McDermott] came on and did well.
"I had a big decision to make this morning, it took me until 10.30am in the shower to decide to put Chris [McPhee] back in.
"I've been umming and ah-ing over it, but it was a big decision because David had been superb the week before against Oxford.
"He responded well in fantastic fashion. I apologised to him for leaving him out after a great performance last week but Chris is an important player.
"Kettering are a big team and I wanted Chris' size and presence but David responded in the right manner and he was big reason why we scored a couple of goals."
It was Courtney who got the ball rolling for Harriers in the second half, opening the scoring only minutes after Kettering midfielder Simon Heslop was sent off for two bookable offences.
It was the defender's first goal for Harriers although something he does often during training.
"It was a good finish by Duane, he does it in training quite often," commented the manager.
"We picked him up from TNS and I knew all about him from when he was at Burnley.
"He had a hard act to follow replacing Keith Lowe and I think he's doing a great job.
Harriers will now go into Tuesday night's league clash at Wrexham (7.45pm kick-off) in confident mood.