Sunday's contest has also been postponed, the Angels announced.
The postponements will give the teams time to continue contact tracing and testing.
Two more players - including outfielder Kyle Garlick - and one staff member tested positive over the last two days, Twins manager Rocco Baldelli said Saturday, according to Fabian Ardaya of The Athletic. The identities of the other player and staff member weren't made public.
Baldelli had originally revealed the staff member's positive test after Friday's game, and he said several people would immediately quarantine at the club's hotel.
The skipper himself nearly missed the road trip after testing positive earlier in the week, although it was later discovered to be a false positive, according to Dan Hayes and Aaron Gleeman of The Athletic.
On Wednesday, the club announced that shortstop Andrelton Simmons tested positive for the virus. However, so far during contact tracing the Twins aren't finding any overlap between their COVID-19 situation and Simmons, sources told Hayes.
Angels manager Joe Maddon expressed some worry about his team being exposed to the virus while playing Minnesota.
"Of course there's a concern," Maddon said postgame on Friday, according to J.P. Hoornstra of the Orange County Register. "There would have to be a concern. ... The fact that we were outside, and we weren't within five feet or whatever the appropriate (distance). ... We'll just let it play out."
Minnesota is currently scheduled to face the Athletics in Oakland starting Monday, while L.A. will host the Texas Rangers. It's not yet known if the positive tests will impact those series.