The British comic told Parade that he and Mirren, who has appeared in more than 50 films, have formed a truly special bond.
"She could get along with anybody, but, yeah, I think we connected with humour," Brand said.
The star continued: "She has a beautiful work ethic and doesn't make a fuss about anything. I've never met anyone more worthy of the title Dame."
Mirren was awarded the title Dame Commander of the Order of the British Empire by Prince Charles in late 2003.
Brand was also questioned as to whether he will retire from stand-up comedy now that he has achieved success as a movie star.
"No," he replied. "I love stand-up more than anything. If you're talking, why not make people laugh?"
Last week, the actor revealed that he frequently sniffed alcohol in order to get into character while filming Arthur.
> Russell Brand 'prefers theme parks to Oscars'
Arthur is released in US cinemas on April 8 and on April 22 in the UK.
Watch the official trailer for Arthur below: