In a statement more than 24 hours after the attack, the group's Aamaq News agency quoted a "security source," saying, "A Unit of Islamic State Fighters carried out London attacks yesterday."
Prime Minister Theresa May on Sunday called for tougher measures to contain Islamic extremism in Britain.
May, facing an election Thursday, said the recent attacks, including an earlier one outside Parliament in March, are not directly connected, but that "terrorism breeds terrorism" and attackers copy one another. She also said five credible plots have been disrupted since March.
"They are bound together by the single evil ideology of Islamist extremism that preaches hatred, sows division, and promotes sectarianism," she said.
The events unfolded Saturday night around 10 p.m. GMT when a van mowed down pedestrians along the London Bridge. Three men fled the van with large knives and attacked people at bars and restaurants in nearby Borough Market, police and witnesses said. The attack unfolded quickly, and police said officers had shot and killed the three attackers within eight minutes.