Hogan, 28, has agreed a four-year contract at St Andrew's after scoring seven goals in 18 appearances during a loan spell last season.
The Republic of Ireland forward moved to Villa from Brentford in 2017 for a fee which it was said could reach £12m.
His most recent goal for the club was in a 3-1 home defeat by Reading in March, their last league game before the coronavirus lockdown was imposed.
After suffering an anterior cruciate knee ligament injury in only his second game, keeping him out for 18 months, he twice had surgery.
But he returned to action to net 21 times in 33 league games, mostly under current Villa boss Dean Smith, to earn his move to the Midlands.