He was stopped by US border officers trying to enter the US from Mexico in 2013 with two containers of liquid.
When he told them they contained apple juice, the agents told him to prove it by taking a swig.
The highly toxic drug killed Mr Velazquez within hours.
His family brought a wrongful death lawsuit against two border protection officers and the US government, alleging that the two agents "coerced and intimidated" Mr Velazquez into drinking the liquid even though they suspected it was a controlled substance.
"I'm not prepared to say they knew for certain that it was going to kill him," the family's lawyer, Eugene Iredale, said. "It's obvious that they suspected from the beginning that it's meth," he added.
Mr Iredale told the Washington Post newspaper that Mr Velazquez had acted wrongly in trying to smuggle the substance across the border "but he's a 16-year-old boy with all the immaturity and bad judgment that might be characteristic of any 16-year-old kid".
The lawyers said Mr Velazquez did not have a criminal record, and it was suspected the boy had been paid a small sum by drug dealers to act as a "mule", carrying the drug across the border.
He tried to cross into the US on foot at the San Ysidro port of entry, near his hometown of Tijuana.
Mr Velazquez started having convulsions soon after taking sips of the liquid and reportedly started shouting in Spanish "my heart, my heart" and "the chemicals!".
He died shortly after being taken to a US medical centre.
Mr Iredale said that the settlement had been reached after several conversations between the parties. Mr Velazquez's parents have been paid the $1m agreed, he said.
The two border officers are still employed by Customs and Border Protection in San Diego.
The family of Cruz Velazquez Acevedo has been awarded $1m (Â£800,000) in settlement after the 16-year-old died from drinking liquid methamphetamine while in US custody.