Cech completed his £10million move to Emirates Stadium on Monday, despite Mourinho making it clear he wanted to keep the goalkeeper at Chelsea.
The Czech Republic international says he realised last season he wanted to leave Stamford Bridge after falling behind Thibaut Courtois in the pecking order, but he has not fallen out with Mourinho over it.
“Jose is one of the best managers in the world, our relationship hasn’t changed although he went with Thibaut Courtois last season.
“We respect each other. He made his decision and I made mine. We had a good chat – I know he wanted me to stay but that’s the way it goes.
“I realised last season how much I love being part of a team week in week out.
“This is not the time for me to be sitting on the bench. It’s a purely footballing reason.
“Roman had the very last word in the end. And I thank him for that.”
Cech has left the Premier League champions for a side that has not won the title since 2004, but the 33-year-old is confident he can continue to enjoy success at Emirates Stadium.
“This was the hardest decision I had to make,” he added. “Last season was a big factor, the manager made his decision – but this time it was my time to make a decision.
“Arsenal matches my ambitions, they want to win trophies. It’s tough for me and some fans but it’s the right decision. The last few days have been overwhelming, the reaction from new team-mates and ex team-mates is unbelievable.
“Last season was difficult. It was a new situation. Match days were difficult, but it’s part of the job of being a second choice keeper.
“I want to win the title, I wouldn’t be sincere if I didn’t admit that.
“Only time will tell if I miss something, can’t forget those 11 years. I look forward to opening a new chapter. It’s a new start, a new motivation.”
Chelsea, meanwhile, are widely reported to be closing on the signing of Stoke goalkeeper Asmir Begovic after making an improved £8million offer for the 28-year-old.