BBC reporter Holly Jones was on the bridge at the time and said a speeding van had veered into pedestrians.
The white van veered off the road on the bridge and hit five or six people on Saturday.
London authorities are investigating a vehicle that hit pedestrians on London Bridge and men with large knives stabbing passersby at a nearby market.
"This is a fast-moving investigation", she said.
An eyewitness who made the emergency calls praised the swift reaction of the police who arrived on the scene in about two minutes.
Shortly afterwards Transport for London said that Vauxhall Underground Station had reopened.
- Terror attack? At 0025 British time on Sunday the incidents at London Bridge and Borough Market were declared as terrorist incidents.
The bridge has been closed at the request of police as a hunt continued for at least three suspects, with police unaware of whether all the terrorists were accounted for.
London police cautioned on Twitter: "We are aware of reports on social media".
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One man who may feel lucky to keep his place is Jason Roy . "It is a really good start and something to build on". He has opened in limited overs cricket at county level, making 174 for Yorkshire against Durham this month.
Another witness, Will Heaven, said he saw people who appeared to have been hit, and one being put into an ambulance.
Senator Dastyari says he was 50 to 100 metres from the iconic London Bridge where a van ploughed into pedestrians, in what is being treated as a terrorist attack.
The attacks, taking place during the Islamic holy month of Ramadan, occurred less than two weeks after a suicide bomber killed 22 people at an Ariana Grande concert in Manchester, and just over two months after a deadly car-and-knife attack at outside the British Parliament.
In March, five people died in London when a vehicle was driven at pedestrians on Westminster Bridge and a police officer was stabbed outside Parliament.
In a related development, some shoppers in the Borough Market, about 80 yards from the London Bridge were injured in a stabbing attack.
Fabio Lamas was working at the Wheatsheaf Pub in Borough Market when he said he heard someone shout "knives, knives, knives" and he saw three people holding knives who appeared to be wearing body armor outside. Police then responded to reports of incidents at two more sites.
Managing Editor of Spectator Magazine was near the London Bridge incident and Tweeted: "What I saw: injured pedestrians, first responders, armed police, people running north".
Although details are scanty, the incident has been confirmed by both the London Metropolitan Police and London Ambulance service via their official handles on Twitter. He thought the person had been drinking, but then police vans flew by.
Nick Archer, who was in the London Bridge area, told Sky News: "We came out (of a bar) on to the road and looked and looked to my left and there as a guy, I thought he was just drinking but he was lying on the floor". Please report anything suspicious to the police, but only call 999 in an emergency. He also suggested that the attack was clearly terrorism and not any kind of accident, calling them "a deliberate and cowardly attack on innocent Londoners and visitors to our city". "There is no justification whatsoever for such barbaric acts", he added. "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.
Explosions heard near the scenes are believed to be controlled police blasts, local media report.