Premier League champions City headed into Sunday's clash with their rivals under huge scrutiny following three winless games.
However, after being helped by Chris Smalling's needless 39th-minute sending-off, Pellegrini's men picked up a fourth successive league derby win for the first time since 1970.
Sergio Aguero grabbed the decisive goal at the Etihad Stadium to leave his manager basking in a "special" win.
"The pressure is the same," Pellegrini told Sky Sports. "We wanted to win the game because we wanted the points.
"We want to retain our title and I think the pressure would have been the same even if we had won recent games.
"It's important always to win the derby because it's special. It is special to beat Manchester United. I am very happy for the three points."
Pellegrini also issued a defiant retort to critics who have suggested City need to tinker with their style of play.
He added: "We have had one or two bad results but I am not going to change the philosophy of what I am trying to do."
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