The action against Pitino came after he was linked to a federal corruption investigation that affected college basketball.
Interim university president Greg Postel announced the promotion of Padgett during a press conference.
"David is known by all as a hard worker," Postel said. "These are the kinds of things that will allow him to be extraordinary in this role."
The 32-year-old Padgett becomes one of the youngest head coaches in Division I basketball. He has been on the Louisville coaching staff for three years, but has been an assistant coach for just one.
Padgett said Postel would not have selected him if he was a target of the FBI investigation.
"I can assure that for myself," said Padgett, who said he has not spoken to the FBI.
He added that two assistant coaches might not be available when preseason practice begins on Sunday.
"We might be a little short-handed here for a while," Padgett said.
Padgett played for Louisville and was a three-year starter at center.
Padgett was Louisville's director of basketball operations for two seasons (2014-16) before he became an assistant coach in 2016.
He becomes just the third head coach for the Cardinals in the last 46 years, following Rick Pitino (2001-17) and Denny Crum (1971-2001).
"It's been a dark week at U of L," Padgett said. "There's nothing else to say about it. Tough week but we're getting through it, putting our head down and getting back to work."