Abbott was handed interim control of the Cumbrians following John Ward's departure last month and the club's former number two has won three, drawn three and lost one of his seven league games in charge.
"I can guarantee everyone who has Carlisle United in their hearts that I will do everything in my power to bring success to the club," he told the official website.
"I've had a lot of support from the players, coaching staff, fans and the board and I know that will continue."
Carlisle's chief executive, David Allen, added: "Everyone at the club is convinced that Greg is the right man to lead Carlisle United forward.
"We had a lot of very good applicants for the job to consider, including many from established and experienced managers, some of who are in good positions with other clubs at the moment.
"But Greg knew he'd be given a fair crack at earning the job himself this time and our thoughts always came back to him."