Bryant, 24, who went 5-for-5 with two doubles and six RBIs, is 10 days younger than Banks was when he hit three homers in 1955.
"I did it in college one time," Bryant said after the game. "I think that's the only time I hit three home runs in a game, and I've never had five hits in a game. I'll remember that for a long time."
Bryant also became the first major leaguer to hit three homers and two doubles in the same game in the modern era, dating to 1900, according to the Elias Sports Bureau.
Bryant hit homer No. 19 on the season in the third inning, a solo shot to left center. In the fourth, he hit a bomb to the upper deck in left field with two men on for No. 20. In the eighth, he hit another solo shot for No. 21. He had six RBIs on the night, which prompted the large contingent of Cubs fans in attendance to ask for a rare road curtain call. Bryant's teammates urged him as well.
"I've never been the type to show up the opponent or anybody," Bryant said. "I don't think that [a road curtain call] has ever happened in baseball, and I didn't want to be the first. I'm glad there were a lot of Cubs fans here, but I'm not that guy."
Cubs manager Joe Maddon wasn't surprised Bryant didn't come out of the dugout.
"He's very old-school," Maddon said. "He doesn't spike the ball in the end zone. He just lays it down, then hands it to the official."
Bryant's five extra-base hits were the most by a Cub since 1885 and his 16 total bases were the most in a game this season, according to ESPN Stats & Information. It was the most total bases since Josh Hamilton's 18 (four home runs, one double) in May 2012.
"I don't know what happened tonight," Bryant said. "The last couple of weeks haven't been what I wanted, so I figured I'm due. ... Throughout my life, I've always been that way, whereas when I'm struggling, I tell myself I'm ready for a big game. I guess tonight was the night."
Bryant said that on Tuesday he'll wear the same ripped pants, which he tore sliding into third base in the first inning, as his teammates kept saying to him, "It's the pants. It's the pants."
He was asked what he'll do with the bat.
"I'm using it tomorrow," he said.
Bryant has 21 home runs and 57 RBIs and leads National League third baseman in All-Star voting. He won the NL Rookie of the Year award last season and could be gaining MVP consideration after his big night.
"He's got that within his abilities," Maddon said.
Bryant is simply grateful that he is better on the road this year, as most of his damage in 2015 came at Wrigley Field. He already has 12 road home runs after hitting five all of last season.
"Last year, I didn't like playing here," Bryant said. "I didn't like playing on the road. This year is a lot different. ... It's definitely the best game of my whole life. I feel so fortunate and so thankful for this opportunity to play on a stage like this every day."
Bryant mashes three homers against Reds
Check out all three home runs from the Cubs' Kris Bryant against the Reds on Monday. The three homers traveled a combined 1,265 feet.