Astros owner Jim Crane announced the dismissals, per Jake Kaplan of The Athletic.
ESPN's Jeff Passan initially reported Luhnow and Hinch were suspended one year each for their roles.
MLB.com's Mark Feinsand reported Astros bench coach Joe Espada, who has been with the team since 2018, will be the team's interim manager. Crane told reporters he will oversee baseball operations for now.
The team is undecided on whether it will hire a full-time manager before 2020 spring training, per Crane (h/t Brian McTaggart of MLB.com).
"I don't have an answer yet, but we'll make it work," the owner noted. He also praised Espada and called him a capable bench coach.
Crane said the team would look inside and outside the organization for GM and manager candidates, per Bob Nightengale of USA Today.
Assistant GM Pete Putila, who Crane referenced in his presser, per Passan, is an internal candidate on Crane's radar.
Putila has been with the Astros since 2011. He was most recently the team's director of player development before the promotion.
The fallout following the MLB's punishments, which also includes a $5 million fine and the forfeiture of the team's first- and second-round draft picks in 2020 and 2021, per Passan, has far-reaching implications outside searching for a new general manager and manager.
Of note, the team's reputation and the integrity of its World Series-winning season have come into serious question.
Commissioner Rob Manfred strongly criticized the team's culture in his statement regarding the investigation:
When asked about those remarks, Crane said the following.
"I don't agree with that," he said, per McTaggart. "We've got a lot of great people. We have over 400 people working here and they work hard. I think there was some isolated situations that led to that."
Still, the Astros "cleaned house," as Crane said, following MLB's decisions. That involved speaking with Hinch to relay his dismissal, and the owner conveyed some moments from that meeting.
"AJ was upset," Crane said. "I think he understands the consequences and said I'm sorry and I'll do whatever I can to help the team."
For his part, Crane also acknowledged his team's guilt.
"We broke the rules, we accept the punishment and we're going to move forward," he told reporters.
The Astros open the 2020 season at home against the Los Angeles Angels on Thursday, March 26.