The attacks come days ahead of a June 8 election and less than two weeks after a suicide bomber killed 22 people at a pop concert by US singer Ariana Grande in Manchester in northern England.
The attack comes just days before Thursday's general election.
London authorities were investigating reports that a vehicle had hit pedestrians on London Bridge.
Khan said the attackers hoped to disrupt democracy and hamper plans for voting, but that would not happen.
Several people said they were ordered to stay inside pubs and restaurants by police and eventually had to come out with their hands on their heads.
A senior Conservative spokesman said national events were being suspended and a decision on further campaigning would be taken later in the day.
A white van hit pedestrians on London Bridge at about 22:00 BST on Saturday, then three men got out and stabbed people in nearby Borough Market.
As they rampaged through the bars around London Bridge, the attackers wore what looked like suicide vests which turned out to be "hoaxes", said Mark Rowley, head of counter-terrorism policing.
Senator Dastyari said he had no details of what had happened but praised the professionalism of police on the scene.
Three men then exited from the vehicle and ran towards Borough Market, where they started stabbing people who were out enjoying the night.
Siti Farhana's family and friends in Malaysia and the United Kingdom have been frantically messaging her to check on her safety since they know she lives very close to the location of the attacks.
After one of the men was shot, Sciotto, captured a photo of the suspect on the ground wearing what appeared to be canisters strapped to his chest.
Media reports said police carried out three controlled explosions during the night. On 22 March, six people died, including the attacker, and at least 50 people were injured in an attack near the Houses of Parliament.
"There was a van that crashed into the fences on London Bridge".
"The suspects were shot dead by armed officers". "He came down the stairs and went to the bar", Dee, 26, who was visibly in shock and declined to give her last name, told AFP.
Simon Thompson told Sky News that he was just outside Borough Market when he saw crowds fleeing.
Ben said: "We saw people running away and then I saw a man in red with a large blade, at a guess 10 inches (25 centimetres) long, stabbing a man, about three times".
Another witness has told Reuters she saw three people with their throats apparently cut on London Bridge.