11:27 AM PT -- TMZ Sports has obtained the arrest warrant affidavit which spells out how investigators were able to identify and catch Vazquez ... and it came down to his distinct tattoos.
According to the documents, Vazquez sent 2 photos and 1 video to the girl -- none of which showed the MLB star's face.
However, officials say the images depicted numerous "unique and distinguishable" tattoos on his body ... which they were able to match to Vazquez through public images on the Internet.
Officials also say the alleged victim's mother was the one who essentially busted Vasquez -- after she discovered the concerning images and video on her daughter's phone.