Tommy, the younger brother of WBC heavyweight champion Tyson, has a long-running feud with Paul, who is the younger brother of YouTuber Logan.
The pair were due to fight over the summer, only for Tommy to have to pull out after being banned from entering the United States.
Tommy was fighting on the undercard of Mayweather's latest money-spinning exhibition fight, and Jake was a surprise co-commentator for the bout.
Jake spent the entire six rounds, of two minutes, insulting Tommy or shouting out to heckle him, prompting a reaction from the latter following the fight.
Tommy shouted abuse back at Jake and was soon backed up by father John, who himself called out the American. "Forget my son, I'm the king of the bareknuckle," he bellowed after taking off his top and shadow-boxing in the ring.
Jake remained outside the ring and simply responded: "Nobody likes you, you're embarrassing. You're a fat, miserable, old man."
Tommy, who won the fight, then demanded Jake enter the ring, to which the YouTuber-turned-boxer instead suggested his rival come to him.
When Tommy attempted to oblige, he was quickly restrained by security as Jack took a few steps back before continuing to mock him on the microphone.