The family said they wanted a more serious, first-degree murder charge as well as the arrest of the other officers involved.
"The pain that the black community feels over this murder and what it reflects about the treatment of black people in America is raw and is spilling out onto streets across America."
The statement called for the city to change its policing, saying: "Today, George Floyd's family is having to explain to his children why their father was executed by police on video."
Shortly after the charges were announced, US Attorney General William Barr said that the justice department and FBI are conducting "an independent investigation to determine whether any federal civil rights laws were violated".
Mr Barr called the video of Mr Floyd's arrest "harrowing to watch and deeply disturbing".
On Thursday, during the third night of protests over Mr Floyd's death, a police station was set alight. A number of buildings have been burned, looted and vandalised in recent days, prompting the activation of the state's National Guard troops.
There have also been demonstrations in other US cities, including New York, Los Angeles, Chicago, Denver, Phoenix and Memphis.
Earlier on Friday, Minnesota Governor Tim Walz said he expected "swift and fair" justice for Mr Floyd's death.
Former President Barack Obama also weighed in, saying: "This shouldn't be 'normal' in 2020 America."
"It can't be 'normal'," added his statement. "If we want our children to grow up in a nation that lives up to its highest ideals, we can and must be better."
President Donald Trump said "thugs" were dishonouring Mr Floyd's memory and called on the National Guard to restore order.
Social media network Twitter accused Mr Trump of glorifying violence in a post that said: "When the looting starts, the shooting starts."
What happened during the arrest?
Officers responding to reports of the use of counterfeit money approached Mr Floyd in his vehicle on Monday.
According to police, he was told to step away from the car, physically resisted officers, and was handcuffed.
Video of the incident does not show how the confrontation started, but a white officer can be seen with his knee on Mr Floyd's neck, pinning him down.
Mr Floyd can be heard saying "please, I can't breathe" and "don't kill me".
Mr Chauvin is accused of causing Mr Floyd's death "by perpetrating an act eminently dangerous to others" and by being negligent, "creating an unreasonable risk and taking a chance of causing death or great bodily harm".
The criminal complaint also accuses Mr Chauvin of having "a depraved mind, without regard for human life".
It says Mr Chauvin had his knee on Mr Floyd's neck for eight minutes and 46 seconds - nearly three minutes of which was after Mr Floyd became non-responsive.
The full report by the county medical examiner has not been released, but the complaint states that the post-mortem examination did not find evidence of "traumatic asphyxia or strangulation".
The medical examiner noted Mr Floyd had underlying heart conditions and the combination of these, "potential intoxicants in his system" and being restrained by the officers "likely contributed to his death".
The Minnesota police handbook states that officers trained on how to compress the neck without applying direct pressure to the airway can use a knee under its use-of-force policy. This is regarded as a non-deadly-force option.