The hosts came from behind to beat the struggling Magpies 2-1 at Brunton Park.
Callum McGregor fired Kiwomya's team ahead in the first half only for Matty Robson to equalise with a clever free-kick. David Amoo's sixth goal of the season settled it after the break leaving County still in search of their first away win
Kiwomya was left to rue missed chances as his side remain stuck in the bottom four.
He said: "I'm really pleased with the way we played and we were the better team by far.
"We must have had at least 20 shots and created numerous chances. But that counts for nothing if you don't finish teams off. We can take confidence with the way we played but we need to work on finishing for the next five days.
"In the last 15 minutes we were pushing on and should have got a goal. I would have been disappointed to get a point, but if we keep playing like that we will get points."
Kiwomya was upset with the decision which led to the free-kick and Carlisle's first half equaliser.
He added: "The ref gave a free-kick for a push but I've looked at it 20 times and still can't understand it.
County defender Alan Sheehan was equally baffled by the outcome after his side had dominated large parts of the game.
"As we were coming off their manager said it was no consolation but we had battered them," said the full-back.