Livesey's poor backpass lead to Lauri Dalla Valle's opener and then the defender blew Carlisle's best chance of an equaliser by blasting over from eight yards.
Crewe dominated the opening 45 minutes finally made the breakthrough two minutes before half-time.
Livesey fluffed his headed backpass and goalkeeper Mark Gillespie blocked Bradden Inman's shot only to watch helplessly as Dalla Valle headed the rebound over him and into the empty net.
Abbott said: "It's an individual error from Danny at both ends that has cost us.
"He's the most honest man in the world and he knows he has made the mistakes, but we are going to get known as great losers if we are not careful.
"There's nothing wrong with the game, the team or the performance, but silly errors mean the best team lost.
"I'm upset and angry as we deserved something from this game. The team's not far away now, but we've given six points away on our travels over Christmas.
"We can't keep giving points away. It's a fine line between being a good side and barging your way up the league or a side that finds life difficult."