The former Brentford striker arrives at Bramall Lane until the end of the season after his departure from Villa Park was sanctioned.
"I've made no secret of my admiration for Scott and his abilities, we made enquiries about him in the last window and we've had to be patient," said boss Chris Wilder.
"He is a player we know a lot about and we are confident that he will be a great addition to what we already have. Options at the top end of the pitch are vitally important and in the situation we find ourselves in, Scott offers us a new dimension.
"Also, Scott had options but he made it clear that he wanted to join us and that goes a long way in my book."
Hogan began his career in non-league, playing for Woodley Sports, Halifax Town, Stocksbridge Park Steels, Ashton United and Hyde before joining Rochdale.
A prolific spell at Brentford then earned him a £12m move to Aston Villa.
Leon Clarke yesterday departed on loan, joining Wigan for the rest of the season.