David Ball and Antoni Sarcevic struck early in each half as Sky Bet League Two side Fleetwood advanced to the Northern Area semi-finals at the expense of two-time winners Carlisle.
The Cumbrians were poor, but should have equalised when space opened up for Brad Potts to sneak into the box, but his shot struck the inside of a post.
Kavanagh said: "It was a very poor performance. It wasn't good enough. In many departments it wasn't good enough.
"I changed it round, I tried to freshen it up. I'll take responsibility, there are players who were given their chance to shine and didn't.
"We are a League One team and they are a League Two team, but it looked the other way round."