Video shows the bus dangling off the overpass. Several people on board were taken to the hospital with injuries.
CBS2's Jenna DeAngelis spoke with one witness, Bryan Garcia, who heard a loud noise from his apartment, which overlooks the scene in the Highbridge section.
"When I see the bus go down the in bridge, I said, 'Oh my God, what happened over there?'" Garcia said.
He rushed down to University Avenue to capture video and saw people yelling for help.
"The ambulance take everybody inside of the bus out,'" he said.
Officials said eight people, including the bus driver, were hurt.
"Preliminary indications are that they bus operator was very helpful, very heroic, was able to help customers get off the bus to safety," said MTA Bus Company President Craig Cipriano.
Jose Luis Brito was one of those customers. He suffered a bruise to his thigh in the accident.
"The bus sped up out of nowhere. It sped up and lost control," Brito said through a translator.
He was on his way home to Hell's Kitchen after visiting his son in the Bronx, CBS2's Andrea Grymes reports.
He says it was something unexpected, that it happened 5-10 minutes after he got on, and he knew something was wrong.
The MTA said the BX-35 bus was headed to a stop around 11 p.m. at 178th and Broadway. It was supposed to make a left turn onto a ramp, when it somehow veered off the road.
As DeAngelis reported, it was an articulated bus with an accordion-style center. It appears the cables in the middle of the bus kept if from fully crashing down.
The FDNY said it was about a 50-foot drop to the road below.
Hours later after 6 a.m., cranes raised the bus back onto the roadway.
"What caused this? Bus drivers are very good with their instincts," neighbor Ilyssa Nichols said. "I've been on these lines for 25 years. I've been in a couple almost-accidents, and their reflexes are good. So for this to happen, there had to be a reason."
The MTA will now do a full investigation to figure that out.
"We'll look at the equipment on the bus. We'll look at whether it had any mechanical issues. We'll look at the event recorder and the video that's on there," said MTA Chief Safety and Security Officer & Acting New York City Transit CEO Patrick Warren said.
CBS2 has learned the 55-year-old driver is still in the hospital with a broken jaw.
MTA officials said he passed a breathalyzer test on the scene, but refused a mandatory drug/alcohol test at the hospital.
"Which is extremely rate for someone not to agree to cooperate with it and obviously extremely troubling," NYC Transit Interim President Sarah Feinberg said.
He's been on the job for 11 years and has a clean record.
Watch Jenna DeAngelis' report --
Officials said the bus was traveling between 17 and 26 miles per hour, when it should have slowed down to 3 to 4 mph to take the turn, officials said.
"While the investigation is ongoing, speed is clearly a factor and is obviously of great concern to us," Warren said.
The vehicle was last inspected on Jan. 13, and officials said there was no indication of mechanical issues during Thursday's crash.
The MTA says the bus has outward and inward facing cameras, plus a black box recorder and GPS, all of which are now part of the investigation.
CBS2 reached out to the bus driver's union but have not heard back.
CBS2's Jenna DeAngelis and Andrea Grymes contributed to this report.
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