Anselmino, who made 10 appearances for Boca in all competitions last season, has signed a seven-year deal but will return to Argentina on loan for the 2024-25 season.
The central defender scored once in their Copa Sudamericana triumph over Trinidense in April, but struggled for much of the campaign with injury.
Anselmino becomes Chelsea's fourth teenage signing of the transfer window following the arrivals of Marc Giui (18), Caleb Wiley (19) and Omari Kellyman (19).
Enzo Maresca has now welcomed eight new faces through the doors of Stamford Bridge, with Atletico Madrid's Samu Omorodion also linked with a move to the Blues.
Chelsea conclude their pre-season preparations this weekend in SW6 against Inter before taking on Manchester City in their opening game of the Premier League season.
The Blues were thrashed 4-1 by the Scottish champions in Indiana, which followed a 2-2 draw with League One side Wrexham in their first match under the Italian.
Chelsea had 10 shots on target in their defeat to Celtic and struck the woodwork twice, but their only goal came from the penalty spot as Christopher Nkunku converted.
They have also struggled in defence, getting caught playing out from the back three times in those games, leaving Maresca with problems to solve ahead of their Premier League opener against Manchester City.
However, Veiga, who played his first minutes in a Chelsea shirt on Saturday, remains optimistic the club will soon see the results of Maresca's tactics.
"Of course, it's about results and in every football game you want to win," he said. "Chelsea especially has to win every game. But we are in pre-season. The most important thing is the process, the connection between each other.
"When the time comes [to play Manchester City] on August 18, that is when it counts and when we have to win for sure.
"He [Maresca] is a very direct manager, very direct in what he wants. You just have to enjoy and learn the maximum that you can in every training session, to give everything you can in every session — that is very important for him as well, and after, you can replicate it on the pitch."
Veiga signed from FC Basel for a reported fee of €14million earlier this month, penning a seven-year contract with the West London club.
The versatile 20-year-old is yet to be told if he will definitely be part of Maresca's plans for the upcoming season, but he is relishing the chance to represent Chelsea.
"How can you say no to Chelsea?" he added. "For me, they are the biggest club in England.
"They have a very ambitious project, not only for the present but the future as well. Right now, the most important thing is the present and I am very happy to be here.
"I am a very versatile player. I just can't wait to get more opportunities to play for this club.
"I was watching them [Chelsea] at home [when I was young], buzzing about Didier Drogba's goals and N'Golo Kante's tackles. I just want to be able to live those moments as well."
