The two teams were scheduled to play Monday through Wednesday in St. Louis.
MLB said Sunday night that the league and the Cardinals "believe it is prudent to conduct additional testing while players and staff are quarantined before the team returns to play."
The Cardinals' three-game series against the Chicago Cubs this weekend was also postponed after the league said two Cardinals players and one staff member tested positive for the coronavirus from samples collected over the previous two days.
The Cardinals announced Sunday that outfielder Lane Thomas also tested positive. In total, 10 Cardinals players and seven staffers have tested positive since last week. St. Louis hasn't played since July 29.
"It's a bummer. We're learning as we go," Cardinals president John Mozeliak told reporters on Sunday night. "What's the right amount of time to not assemble to try and prevent the spread? I don't think we know that answer perfectly."
Mozeliak added: "We're doubling down on our tracing."
There have now been 27 games postponed by Major League Baseball because of coronavirus concerns. The Cardinals have had 13 games scrapped.
ESPN's Jesse Rogers and The Associated Press contributed to this report.