Shakespeare was quick to praise Wood's all-round contribution to the side after Leicester ran riot with a four-goal burst in 16 minutes after half-time.
Carlisle had taken an early lead through David Amoo, but Wood levelled before the break and added two more in the second half, one of them a penalty. Lloyd Dyer and Anthony Knockaert were also on target before James Berrett had the final word for the well beaten Cumbrians.
Shakespeare said: "It will do Chris' confidence the world of good. He is a team player and his finishing was very clinical.
"You could see at times that Carlisle are a struggling side, but we are a good team when we pass and move and they won't come up against that very often in the league they are in.
"We probably should have scored a few more, but the end result was very pleasing. When you go behind the crowd get up, but we managed the ball well and opened them up on many occasions."
Hat-trick man Wood added: "Carlisle did their homework and closed us down early on. It's nice to be off the mark which was what I was looking to do. We are in the next round which is the main thing.
"If we had been more clinical we could have had more goals and their 'keeper did well to keep them in it."