The win, Carlisle's first in four matches, took them up to 16th in the table and Abbott insists the club are working toward a long-term plan.
"I'm not the type who gloats in victory or gets too upset in defeat. I keep a calm head but under a lot of pressure my players have responded," he said.
"The dressing room was amazing before the game and the team is so together and so united that there is no way we will be stuck in this position for long. We have a long-term plan and we're going to be patient and do it together as a team.
"Crewe have been beaten by a good side and I hope they hold their hands up and admit that."