Chelsea manager Jose Mourinho was a major factor in persuading Pedro to join the Premier League champions.
The former Barcelona winger, who was a target of Manchester United, made the £21m move from Spain on Thursday.
"He phoned me a few times and said he needed me to make the team stronger," the 28-year-old told Chelsea.com.
"He has won titles wherever he has been. I am really looking forward to playing under him and hopefully winning more titles."
The Spain international, who won three Champions League and five La Liga titles with Barcelona, trained with his new team-mates on Thursday and could make his debut in Sunday's match at West Brom.
The move reunites him with former Barcelona midfielder and Spain team-mate Cesc Fabregas.
"I spoke to Cesc," said Pedro. "We are very good friends. I knew I would feel at home here and that was one of the main reasons why I decided to join.
|More from BBC Sport|
|Why Pedro should be a good match for Chelsea|
|The Man City youngster 'living the dream'|
|Where does Pedro 'snub' leave Man Utd?|
|Get Inspired: Bowling tips from England's Woakes|
"He's a great player, he's got brilliant vision and he links up really well with the other attacking players. I know we work well together on the pitch so I can't wait to play with him again.
"I have also played with Diego Costa and Cesar Azpilicueta before, for the national team, so I feel very comfortable here."
Pedro had been a target for United throughout the summer, with the club's executive vice-chairman Ed Woodward flying to Catalonia on Monday in an attempt to seal a deal for the player.
But things changed on Wednesday as it emerged that United boss Louis van Gaal had withdrawn his interest while Chelsea were moving to sign him ahead of their rivals.
"I am ready to start playing straightaway if the manager selects me," he said.
"I have helped Barcelona win titles and I want to do exactly the same thing at Chelsea, because ultimately we are all here to win things.
"It's a new challenge and I can't wait to get going."