Burton beat the Foxes 4-2 at the King Power Stadium back in August but it was a different story this time, Chris Wood and Ritchie De Laet netting first-half goals as the hosts eased through.
Shakespeare said: "We got our fingers burned against Burton the last time and they deserved it on the day, 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."
Burton manager and former Leicester player Gary Rowett said: "It was the not the greatest start, conceding a goal after three minutes.
"In the week we had worked on having a very solid base because we know the sort of side Leicester are.
"They are going to come at you, and at home they are a terrific side. Their home form is sensational this season.
"It was important that we weathered that early storm, so to concede after only three minutes leaves you with an absolute mountain to climb.
"You can't go stupidly forward because you then risk conceding two or three more goals and, when the second goal went in, it makes it that much harder.
"To hope to do it twice at Leicester was always going to be difficult. You look at Leicester's team and think they could get five or six if you go at them. They are a terrific side and I hope they can get up into the Premier League."