Championship Wednesday threatened a famous FA Cup upset in Sunday's third round tie at the Etihad Stadium when Atdhe Nuhiu stroked the ball home from the edge of the box in the 15th minute.
However, Premier League champions City improved after the break and birthday boy James Milner finished off a fine team goal in the 66th minute, before the England international poked home an injury-time winner.
Manager Pellegrini admitted his team fell well below their usual standards, although was quick to mitigate that with a congested fixture list that has seen City play four matches in 10 days.
"We did not play well," Pellegrini told BBC Sport. "It was our fourth game in 10 days so I think we were feeling a little bit tired and I think when you are playing a team from a different division the motivation is different.
"We played very badly, we didn't keep the ball and we didn't play the way we normally do.
"One of the most pleasing things about the way we have played in the last month is that we have continued winning, even without any recognised strikers, but playing the way we have, we've still been able to score goals.
"That's why today I found it hard to believe why we weren't creating enough chances."
Pellegrini is now unconcerned with who City draw in round four.
"It doesn't matter as the ties are going to be tough whoever we face," he added.
"It's a very important competition for us, but we have to take it a game at a time and we will try to stay involved as long as possible."
