Skrill South side Wood had chances to win the first-round tie through Graeme Montgomery and David Moli, but Sean O'Hanlon hit the post for Carlisle.
Kavanagh said: "I thought a replay was fair.
"It was certainly a battle. I didn't think we performed anywhere near the level we can do and certainly not anywhere near the level we produced on Tuesday night against Wolves. But that's what happens in the FA Cup.
"Disappointing that we now need to have an extra game, but pleased to still be in the Cup.
"It was their Cup final and they were always going to compete with us in every department and make life as difficult for us as they could.
"In truth we felt we made life difficult for ourselves."