The move means David Bote will be the opening-day second baseman Thursday against Pittsburgh.
"This is less about Nico and more about David Bote," Cubs manager David Ross said. "Bote had a great spring. It's an opportunity for David that he's earned and deserves. It's a chance to play every day at second base.
"I dont think it's fair for Nico to come off the bench and have sporadic at-bats. That's not setting him up for success. And I think David Bote earned the job."
Bote, who turns 28 on April 7, batted .200 with seven homers and 29 RBI during the pandemic-shortened season last year. He was selected by Chicago in the 18th round of the 2012 amateur draft.
Infielder Eric Sogard, 34, will also be added to the big league roster after entering camp on a minor league deal.
Hoerner, a first-round pick in the 2018 amateur draft out of Stanford, also had a strong spring, batting .361 with a homer and six RBI.
The Cubs could delay Hoerner's eligibility for salary arbitration until after the 2023 season if they keep him in the minors for 30 to 50 days, unless there are changes in the next collective bargaining agreement. He batted .222 with no homers and 13 RBI last year during the pandemic-shortened season.
Williams, who signed a $2.5 million, one-year deal with Chicago in free agency, is looking to rebound after he went 2-8 with a 6.18 ERA in 11 starts with Pittsburgh last year. Alzolay, a rookie right-hander, had a 1-1 record and a 2.95 ERA in six appearances with the Cubs last season.
Alec Mills, who threw a no-hitter last year at Milwaukee, will work out of the bullpen for the reigning NL Central champions.
Veteran outfielder Cameron Maybin was released. Right-handers Shelby Miller, Pedro Strop and Trevor Megill and outfielder Rafael Ortega -- all non-roster invitees -- were assigned to minor league camp.
The 30-year-old Miller had pushed for a spot on Chicago's staff, either as a starter or a swingman out of the bullpen. He allowed one earned run in nine innings this spring, striking out 12.
The moves reduced the Cubs' spring roster to 31 players.
ESPN's Jesse Rogers and The Associated Press contributed to this report.