The post-apocalyptic first-person shooter was developed by Bungie. It was the US studio's first release after quitting the Halo series.
The title attracted mixed reviews, and its relatively bare-bones plot continues to divide gamers.
The award surprised many in the audience at the London ceremony as it had not won any of the other categories it had been nominated for.
However, Bungie's skill at crafting alien-slaying gunfights appears to have helped it seize the top prize.BBC Newsbeat's Jonathan Blake tweeted the reaction to Destiny's victory from the ceremony
Accepting the award Bungie's president thanked Destiny's players.
"It's always a labour of love that we do because of the passion of the fans," said Harold Ryan.
"They really are the ones who drive you to put the energy and time into putting the game together."
Several other biggest-budget nominees - including Assassin's Creed Unity, Mario Kart 8 and Call Of Duty: Advanced Warfare - walked away without a prize.Rufus Hound hosted the British Academy Games Awards for the first time Big name losers
In fact, many of the winners managed to secure what might be seen as David-and-Goliath battles.Ashley Johnson now has two Baftas for playing Ellie - she won a prize for the same character last year
OlliOlii - the low-budget skateboarding video game from London-based studio Roll7 - beat both Fifa 15 and Forza Horizon 2 for the best sport video game.
The actress Ashley Johnson defeated the Hollywood star Kevin Spacey to win the best performance award for Ellie in The Last of Us: Left Behind.
And the puzzle game Lumino City - which was created using real-world models made out of paper, card, miniature lights and motors - beat Ubisoft's Far Cry 4 and Assassin's Creed Unity for artistic achievement.
Monument Valley won British game of the year. Its developers, UsTwo Games sparked a brief backlash in November after charging ?1.49 for extra levels - a fee many gamers thought justified after trying them out.
The title also took the mobile and handheld category.
Alien: Isolation, by Sussex-based The Creative Assembly, only walked away with the audio achievement prize despite entering the ceremony as the favourite with six nominations.Return winner
The awards were hosted by the comedian Rufus Hound - his first time in the role.
His opening monologue poked fun at the rising number of video games that sell expensive downloadable content in the months following the original title's release.
However, that did not stop last year's big winner, The Last of Us, from also scooping up the best story award for its add-on tale, Left Behind.Analysis: Adam Rosser, presenter of Radio 5 live's Game On
There was an audible ripple of surprise in the press room as Destiny took the best game Bafta.
Destiny has been criticised in many quarters for being the epitome of "grind", suffering from a sparsely populated game world and repetitive gameplay.
In many people's minds the game has struggled to integrate two genres: the first-person shooter and the massively multiplayer online (MMO) game.
That was always going to be a tall order.
As was following the well-thought of Halo series, which Bungie built its reputation on.While some have suggested that "grinding" through Destiny's battles to get better gear becomes boring, others say its fights represent a pinnacle in gaming combat
Destiny took the template that Halo established and spun a similar aesthetic out into a different kind of universe.Destiny has its defenders, including The Verge's senior editor Tom Warren
One in which Earth is threatened by "the Darkness", a nebulous adversary that wouldn't be out of place in a Luc Besson film, and the central "character" in the game, The Watcher is a mute orb in whose shadow the last safe city on Earth is founded.
There are good ideas scattered through the game but at present, several DLC (downloadable content) releases in, it still manages to feel unfinished.
In a strong category - up against Alien: Isolation, Middle Earth: Shadow of Mordor and even Mario Kart 8 - the decision to give the most high profile award to Destiny looks, to many, like an eccentric one.
Listen to Game On's weekly podcastPaying respects
The awards this year introduced a new category - best persistent game, representing titles whose action continues even when the player is not taking part.
Destiny had been nominated for the prize, but it went instead to Riot Games' hugely popular League of Legends.League of Legends became the first title to win the award in the Persistent game category
Following the announcement, Mr Hound paused to pay tribute to the fantasy author Terry Pratchett, whose death had been announced hours earlier, and the late Star Trek actor Leonard Nimoy, who voiced several video games.
There was also a sombre moment following the award for "original property" when the team behind Ubisoft Studio's Valiant Hearts paid their respects to the Word War I soldiers who had inspired its puzzle adventure.West Sussex-based Alien: Isolation only took one award for audio achievement
This year's winner in the "ones to watch" category was Overly Kinetic, a development team made up of students from the University of Southern California.
They created the distinctive "stealth-deathmatch" multiplayer game Chambara, in which four players camouflage themselves in the game's environment while trying to sneak up on and attack each other.
The category is linked to the Dare to be Digital competition run by Abertay University, which gives contestants nine weeks to create a prototype, and is intended to bring attention to new talent.The five-person team behind the highly stylised multiplayer stealth-action title Chambara took the ones to watch award
Another American team, albeit a more experienced one, celebrated taking the "multiplayer game of the year" prize - Blizzard Entertainment for Hearthstone: Heroes of Warcraft.
The developer beat Destiny for the prize.
The creators of the strategy card game revealed during their acceptance speech that some in their studio were dubious about its prospects before its release.
The title - which is a spin-off from the role-playing game World of Warcraft - recently revealed it had attracted 25 million registered players.Monument Valley won the British video game award but missed out on the top category Fellowship
David Braben - the creator of Elite and its most recent sequel Elite: Dangerous - received a standing ovation when he took to the stage to be given the Bafta fellowship by another British gaming legend, Ian Livingstone.
Mr Braben suggested that we are on the "cusp of a golden age for video gaming".
"The rise of the indies is truly amazing," he said. "It's the best time there has been to be in this wonderful industry."
He thanked members of the public who had crowdfunded his latest title and his colleagues at Frontier Developments.
At one point it had seemed that Elite: Dangerous would not raise its target of ?1.25m on Kickstarter, which would have meant it would have missed out on the seed money.
Microsoft recently revealed that the title would soon be released for its Xbox One console following its success on PC.Category Winner
Monument Valley (UsTwo)
Never Alone (Upper One Games)
The Last of Us: Left Behind (Naughty Dog)
Ashley Johnson for The Last of Us: Left Behind
Lumino City (State of Play)
Alien Isolation (The Creative Assembly)
Minecraft: Console Editions (Mojang/4J Studios)
Middle-earth Shadow of Mordor (Monolith Productions)
The Vanishing of Ethan Carter (The Astronauts)
Mobile and handheld game
Monument Valley (UsTwo)
Hearthstone: Heroes of Warcraft (Blizzard Entertainment)
Far Cry 4 (Cliff Martinez, Tony Gronick and Jerome Angelot)
Valiant Hearts (Ubisoft Studios)
League of Legends (Riot Games)
Ones to watch
Chambara (Overly Kinetic)
David Braben (Frontier Developments)