Freddie Sears' equaliser denied the Cumbrians victory, after Matty Robson had given them a fourth-minute lead, but Abbott's team are off the mark after losing all three previous league matches.
"We're a little bit prouder and a little bit happier," said Abbott. "We had our best players back available and it showed on the counter attack.
"With what we'd gone through, we needed to make the start we did and to get some encouragement.
"Colchester have had a fantastic start to their season and we've watched them at Sheffield United, where they probably deserved to win.
"They've had a great start whereas we've had a disastrous one, which is probably being kind to it.
"It was about fighting together and working together and making life as difficult as possible for the opposition.
"If at the end of the day we'd lost because of that, then you have to hold your hands up. But we worked exceptionally hard and we got our rewards for that.
"I thought it was a fantastic game which was end to end and we contributed to a decent football match."
Colchester made it four games unbeaten in League One and manager Joe Dunne said: "It was end-to-end stuff in the first half but one team dominated in the second half and that was us.
"I was delighted with it and we kept patient and kept passing it. We scored a good goal and we did everything we could do to win the match.
"In the second half, we were absolutely outstanding. We were a bit sloppy at times in the second half with some square passes but what really pleased me was the patience of the players.
"Carlisle started well but I was really disappointed with the way that we started the game. But I haven't got a bad word to say about my players.
"We said at half-time that if we kept passing then we would get opportunities.
"The players are disappointed that they were unable to win the game and we're due a big result with the chances we create."