The home side scored with their only shot on target - a fantastic finish by the hard-working Alan Judge in the first half - to close the gap on the top six.
However, County were grateful to a stunning display by Bialkowski who pulled off a string of terrific saves to deny Carlisle.
Abbott said: "We asked for a performance and we got one. It was a terrific performance and I've got absolutely no qualms about that.
"The way they battled, the way they competed and the way they played football on a cabbage patch was absolutely terrific.
"We just couldn't beat their man of the match, the goalkeeper, but that happens in football and we've got to be big enough and ugly enough to take it.
"We've lost the game to an inspired goalkeeper and because we've not had the luck around the goal. You can't be too critical and I won't be."
Abbot was also left disappointed as he faces three matches without his skipper Lee Miller after the striker was sent off for an alleged headbutt late on.
He said: "I had turned my back so I didn't see it but Lee is adamant he didn't headbutt him. There is no blood and when you headbutt someone you expect blood so maybe it wasn't a severe headbutt and no-one has gone to hospital."