The record follows a break during which singer Billie Joe Armstrong went to rehab for prescription pill addiction and touring guitarist Jason White received treatment for tonsil cancer.
"To have a chart-topping album at this stage of our career is especially gratifying," the band said.
As with 2004's American Idiot, the record is a critique of US society.
The album's first single, Bang Bang, addresses the culture of mass shootings, telling the story of one such incident through the eyes of a perpetrator.
The title track was written after Armstrong joined a march in Manhattan, New York, protesting against a grand jury's decision not to indict a white police officer for the killing of unarmed black teenager Michael Brown in Ferguson.
"It was happening all over the country," he told Rolling Stone magazine. "It was a trip to see people rebel against the old order.
"I was feeling that people don't want to feel obsolete in the things that we care about."
Revolution Radio outsold its nearest competitor, Barry Gibb's In The Now, by a ratio of two to one, said the Official Charts Company.
Other new entries in this week's album chart came from US rock band Alter Bridge, who landed at three with The Last Hero, and the Kaiser Chiefs, whose pop-centric album Stay Together debuted at four.
Pop band OneRepublic scored a new entry at six with Oh My My, while Norah Jones entered at 10 with Day Breaks, a record that marks a return to the jazz-inflected style of her debut, Come Away With Me.
In the singles chart, James Arthur holds on to the number one spot for the third week in a row, with his downbeat ballad Say You Won't Let Go.
The singer strengthened his lead over the competition thanks to a performance on last week's X Factor, which resulted in 50,000 sales and 5.16 million streams - the highest figure his song has achieved so far.
Bruno Mars was the highest new entry 24K Magic, a charismatic retread of his hit single Uptown Funk; while Swedish star Zara Larsson edged closer to the top 10 with her single Ain't My Fault rising three places to number 13.
Actress-turned-pop star Hailee Steinfeld had the highest-climbing single of the week, as Starving - her collaboration with dance producer Zedd - leapt from 28 to 17.
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Punk-pop band Green Day have hit number one for the third time in their career with their new album Revolution Radio.