Villa's seventh straight league win saw Steve Bruce's side climb to second place, 12 points behind runaway leaders Wolverhampton Wanderers.
Adomah opened the scoring on the hour with his 13th league goal of the season before Hourihane volleyed his side's second.
Birmingham's Cheikh Ndoye was sent off in stoppage time for a second bookable offence.
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Villa jumped ahead of Derby into the second automatic promotion spot as they look to return to the top flight after being relegated from in 2016.
(Reporting by Martyn Herman; Editing by Toby Davis)
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