Johnson was among the scorers as Roberto Mancini's side opened up a 3-1 lead at the half way stage, but Lloyd Dyer scored to set up a nervy finish before Aleksandar Kolarov sealed the win in the closing stages.
"They pushed us, and in one-off games anything can happen," Johnson told ESPN. "But we made it hard for ourselves. I didn't think it was a penalty for their first goal, and the ref played a one-two for their goal and otherwise it would be 4-0.
"When we go two goals up here we shouldn't be put under pressure but a little break for them gets them back in the game, but Kolarov scored a great goal at the end."
Mancini was pleased his team progressed but he would like to see City improve their defensive form, which has seen them ship ten goals in their past three games.
"It is always difficult in the FA Cup," he said. "Leicester played very well in the first 20 minutes but after we played better and scored the three goals.
" But again in the last 15 minutes, like against Wolves, we thought that it was finished and we conceded another one. We must improve. We have conceded a lot of goals in the last three games, so we must be sure to work hard.
Leicester boss Sven-Goran Eriksson made his return to the City of Manchester Stadium and he was satisfied with the Foxes' efforts.
"We were in the game for 90 minutes at home, and 87 here, and that's good," the former England boss said. "They were better than we were tonight, but we never gave up and from 3-1 down at half-time, coming back to 3-2, we're in the game again and of course I'm happy with that.
"It's a little difficult. They have some better players than us, but I think we need to be proud.
Eriksson now wants his club to focus on climbing the Championship table.
"Maybe there will be some different type of games, but we have shown in these two games we can play good football and stand up to physically strong teams. The rest of the season is very much about that."