Betts had already performed the feat on May 31 in Baltimore before drilling three baseballs over the Green Monster in the first five innings on Sunday.
Williams, one of the most accomplished hitters in history, had three-homer games just over a month apart in 1957.
"It makes you feel pretty good," Betts said. "Any time your name is said with his, you know it means you've done something well. He had a great career, and mine's just starting."
The 23-year-old Betts is one of the game's rising stars, and he is hitting his way into the American League's Most Valuable Player Award discussion.
Following his 4-for-6 display on Sunday, Betts had a slash line of .313/.353./.561 with 91 runs scored, 34 doubles, five triples, 26 homers and 84 RBIs.
The production numbers are particularly impressive when you consider he spent most of the season batting leadoff before manager John Farrell moved him to third earlier this week.
"When we made this move, there was some thought that if Mookie's going to stay on the power run that he's on, maybe it's got a chance to be with some people on base," Farrell said.
Betts is now tied with Ortiz with the team lead in homers and he passed his teammate for the Major League lead in extra-base hits with 65.
"Yeah, it's pretty spectacular when you see the first couple of innings," Farrell said. "This is the third time this year he's had home runs in the first couple innings of the ballgame. Second time he's hit three home runs in a game. Just electric bat speed against a very good pitcher in [Zack] Greinke. Just one heck of a day for Mookie."
The eight-RBI performance was the most by a Red Sox player since Bill Mueller on July 29, 2003.
As the day unfolded, the only question was whether Betts could tie the Major League record with four homers in a game.
His teammates told him to swing for the fences with the game not in doubt.
"I mean, they told me, 'Try and swing for it.' The first couple of times, I didn't," Betts said. "If it was going to happen, it was going to happen. I was just trying to get another and try and make it a five-hit day like [Dustin Pedroia]."
What excites Betts the most is that he's playing in a pennant race for the first time in his career after being part of last-place teams the past two seasons.
MVP? He'll let the voters worry about that.
"You know, we play the game to win a World Series, so I'm just trying to do that," Betts said. "All the other stuff can come and go, whatever, but my main focus right now is just to make the playoffs and win a World Series."