James Milner struck twice, including a stoppage-time winner, as the Barclays Premier League champions came from behind to beat their battling Bet Championship opponents 2-1 at the Etihad Stadium.
City toiled after falling behind to an Atdhe Nuhiu strike in the 14th minute and only roused themselves in the latter stages with Milner switching to a forward role and equalising after 65 minutes.
Pellegrini said: “Yes I am relieved because we didn’t play well.
“We play very badly except the last 20 minutes, when we hit another gear to try to create the space against a team that defended very well.
“When you play against this kind of team it is difficult when you are 1-0 down.”
Milner’s double was the perfect way for the England international to celebrate his 29th birthday, but his future at the club is still to be resolved.
Milner’s contract expires at the end of the season and, although talks over a new deal have reached an advanced stage, nothing has yet been signed and he has been linked with Liverpool.
Pellegrini said: “(There is) no news. I hope we will find an arrangement.
“But I think that I have said already what I think about James Milner. He is very useful. He always plays with 100 per cent commitment and intensity.
“I am very happy for him and also as it his birthday.”
Milner grabbed the headlines but the real catalyst for City’s fightback was the double introduction of playmakers David Silva and Samir Nasri just after the hour.
Pellegrini had rested the pair among six changes but they made an immediate impact after coming on. They both had a hand in the equaliser and Silva had a chance to win the game himself late on before setting up Milner’s second.
Pellegrini said: “We have a lot of players and (have had) a lot of games in the last 10 days. We have played four.
“We didn’t do well but we when we returned to our normal offensive formation with (Jesus) Navas, Silva, Nasri and Milner on top we scored two goals.”
Wednesday’s performance was much improved from their last visit to Eastlands, when they were thrashed 7-0 in the Capital One Cup in September.
Manager Stuart Gray was pleased with the Owls’ performance and felt just one late lapse cost them.
Gray said: “Last time we were here we got beat 7-0. It was 0-0 at half-time and they went up a gear and put seven past us.
“It was nice to go in at 1-0, I think we thoroughly deserved it. We had a gameplan to hit them on the break, we frustrated them.
“The goal that we got – Stevie May pulled away on the break and pulled it back to Atdhe Nuhiu and it was a great finish.
“In the end they have shown us total respect as they put on Silva and Nasri and (Gael) Clichy. It was just tired minds that caught us out.
“They kept the ball better than us in the last 15 minutes and when you have tired minds, you can make mistakes unfortunately. The winning goal was a mistake by us but I am so proud of my players. They have pushed the champions to the last minute.
“I thought it was going to go to a replay. It just shows in this game that you have to play until the final whistle.”