The album, "Hackney Diamonds", is the first since the death of drummer Charlie Watts in 2019, and the first containing original material since 2005's "A Bigger Bang".
"We didn't want to make just any record and put it out," Jagger, 80, told a launch event at the Hackney Empire theatre in east London, hosted by US talk show host Jimmy Fallon and livestreamed on YouTube.
"I'm not saying we're big-headed about it but we're pleased with it and we hope you all like it," the octogenarian rocker said.
The Stones' 24th studio album, set for release on October 20, has 12 tracks, with the single "Angry" the first to come out.
Wood, 76, confirmed during the launch that superstar Lady Gaga features on another, "Sweet Sound of Heaven", adding she "sings really sweet" on it.
Two of the tracks were recorded with Watts in 2019. The others feature Steve Jordan, whom Watts recommended to replace him.
"Ever since Charlie's gone it's different," said 79-year-old guitarist Keith Richards of recording the new album without one of the band's founders.
"He's number four, he's missing, he's up there. Of course he's missed incredibly."
- Already acclaimed -
"Hackney diamonds" is English slang for the shards of glass left scattered on the ground after smash-and-grab robberies, and refers to the historically working class east London neighbourhood.
"It's like when you get your windscreen broken on a Saturday night in Hackney," Jagger joked.
Throngs of people crowded outside the event venue, including those without tickets who came out of curiosity and hoping to spot the band.
"I've been following the Stones since I was four years old and I'm from round here -- it's my backyard," said musician and fan Rory McGlinchey.
"It's crazy that they're here!"
Wearing a Rolling Stones t-shirt, he said the new release was "great news". "Can't wait," he told AFP.
Critics have already hailed the release as the band's best work in many decades.
Will Hodgkinson, rock and pop critic for Britain's The Times newspaper, said it is "unquestionably the best Stones album since 1978's 'Some Girls'".
"Variously poignant, irreverent, anarchic and, in one gospel-tinged moment, quite spiritual, it touches on all the aspects we love about the band, glued together by the rambunctious energy they have made their own since the early Sixties," he wrote.
The Daily Telegraph's music reviewer Neil McCormick said "Angry" was "their best single in four decades".
The Stones also unveiled the single's typically rock'n'roll video.
It sees the band playing from Los Angeles' billboards as scantily-dressed, leather-clad "Euphoria" actress Sydney Sweeney writhes to the soundtrack in the back of a convertible red Mercedes driving through the city.
- Teaser -
The Rolling Stones' last studio album was 2016's "Blue & Lonesome", which was made up of blues covers.
"We've been very lazy," said Jagger. "We've been on the road most of the time."
The band teased the release of the new album through a spoof advert in the local Hackney Gazette newspaper.
The cryptic ad, which also appeared in sister title the Islington Gazette, referenced several of the band's best-known songs including "(I Can't Get No) Satisfaction", "Gimme Shelter" and "Shattered".
The ad went on: "Opening our new store on Mare Street, September 2023. Our friendly team promises you satisfaction when you say gimme shelter we'll fix your shattered windows."
Clues that the ad was not for a bona fide east London glass repair business included a miniature version of the band's famed lips logo to dot the letter i.
The ad -- in the same font as the "Some Girls" album -- also says the firm was established in 1962, the same year the band was formed.
Last year the Stones travelled through Europe for their 60th anniversary tour which featured stops in cities including Madrid, Milan and Munich, and also a performance at British Summer Time (BST) festival in London.
Asked for the secret to their decades-spanning musical marriage, Jagger quipped: "not speaking too often".
"How to say shut up politely," added Richards.