Tests showed that Bieber was under the influence of drugs, Miami Beach Police Chief Raymond Martinez told the newspaper.
The other driver, identified as Khalil Sharief, was charged with drink driving, police spokeswoman Vivian Hernandez said.
Both were taken to Miami-Dade jail, where they were fingerprinted and had their mugshots taken.
"Miami nights ridin thru yo city in that hot wheel," Sharief, an R&B singer, tweeted on his account earlier on Thursday morning.
Bieber, 19, was incoherent, had his hands in his pockets and resisted arrest without violence, Martinez said. He also did not have a valid licence, the chief added.
"It was a cold night, and it was quiet, so the officers could hear the cars," he said, adding that patrols followed the sound of the revving engines.
"When the cars are going that fast, it takes a while to catch up."
Bieber told officers that he had a beer, was under the influence of anti-depressants and had been smoking marijuana all day, according to police sources.
Officers at the Miami Beach police station worked for hours on booking Bieber into jail.
The arrest caps a highly visible week for Bieber in South Florida, where he is reported to have blown thousands of dollars at the King of Diamonds strip club.
Also, the Opa-locka Police Department has said it is investigating officers who gave Bieber and his crew an unauthorised escort through traffic after landing at the Opa-locka airport.
Two officers have been relieved of duty with pay, pending the outcome of the probe, sources say.
His arrest is another black eye for the embattled Boyfriend singer, who has had a fondness for drag racing in residential neighbourhoods.
His neighbours in his exclusive, gated Calabasas, California, community have long complained about him speeding down their streets.
Earlier this month, a neighbour accused Bieber and his pals of hurling eggs at the neighbour's home, causing extensive damages.
Officers searched Bieber's home and one person was arrested for drugs. He is under investigation for vandalism in that incident.
- MCT