League Two Burton beat City 4-2 in August, but were never in the hunt in the third-round FA Cup tie with first-half goals from Chris Wood and Ritchie De Laet giving Leicester a 2-0 win.
After scoring two goals on his debut against Huddersfield on New Year's Day, Wood snapped up another in the third minute, prodding the ball into the net after a through pass from Paul Konchesky.
It was the perfect start for the Foxes, and they added a second goal in the 21st minute. Lloyd Dyer's shot was blocked, but De Laet,who scored his first goal for the club in the 6-1 win against Huddersfield fired in the rebound. The second half fizzled out as Leicester eased through to a comfortable victory with Burton offering little threat.
"We got our fingers burned against Burton the last time, and they deserved it on the day," said Shakespeare.
"So we didn't want a repeat of that. I wouldn't say that we came through without a scare because they always say that 2-0 is a dodgy scoreline.
"But the first goal was very important for us and after that the boys went about their jobs in a very professional way. We restricted Burton to very few opportunities but we knew what they were capable of.
"It was also good to keep a clean sheet. Our team selection showed respect for Burton and we also wanted to keep momentum going and give some of the lads game time.
"The league is our priority but modern players can play two games a week and the FA Cup is special.
"There is nothing wrong with having a good cup run. We knew it was going to be a tricky tie so we are pleased to be through to the next round."