The 26-year-old scored 20 goals in 52 games for Bristol City, after signing from French club Angers in July 2015.
The Ivory Coast international had also been the subject of interest from Derby County and Premier League side Hull.
Villa had reached an agreement to sign Hull forward Abel Hernandez, but the move fell through over personal terms.
Bristol City owner Steve Lansdown had previously said they would not sell Kodjia to any fellow Championship club, and that any deal would have to exceed £10m.
The deal represents the largest transfer fee City have ever received for a player, surpassing the £3.5m the club sold Ade Akinbiyi to Wolves for in September 1999.
Villa boss Roberto Di Matteo said he had signed a "proven goalscorer" in Kodjia, while City chief operating officer Mark Ashton said the club had retained a sell-on clause and decided to disclose the fee "due to the high-profile nature and speculation surrounding the deal".
Ashton added: "We've continually told Jonathan that we don't want him to go, but recently he's made it clear that he's aware of the interest of other clubs and that he wanted to leave."
Kodjia is a proven goal scorer at Championship level, a physical and competitive style of player with greater mobility than a Rudy Gestede or Libor Kozak.
Whether he is quite the "world-class striker" that Villa chairman Dr Tony Xia promised via his prolific Twitter account is dubious, but he could well thrive on the sort of service Ross McCormack and Jack Grealish are capable of providing.
The Villa squad was always going to need revolution rather than evolution after last season's psychologically damaging relegation - Kodjia is the latest component of a summer of change.

Sky Mobile will offer flexible monthly payment plans and the ability to roll over unused data each month for up to three years.
Sky will become a "quad play" provider, offering broadband, television fixed-line telephone and a mobile service.
And existing Sky TV customers will not have to pay for calls or texts and can compile playlists of their favourite shows to watch on their mobiles.
According to Sky, some 46,000 have pre-registered for the service.
The service comes with three data plans:
Non-Sky TV customers can add on unlimited calls and texts for £10 per month, or pay for calls and texts on a pay-as-you-use (PAYU) basis.
According to Sky, UK consumers use less than half of the mobile data they buy from mobile network operators, adding up to an annual £2bn of wasted data.
Stephen van Rooyen, UK and Ireland Sky chief executive, said:  "We felt it was time to shake up the mobile market and give customers a completely new way to manage their mobile plan.
"We've designed it based on what people told us they want - it's easy, flexible and transparent and it puts the customer in control."
Ewan Taylor-Gibson, a telecoms expert at uSwitch said: "Sky's first mobile offering is very competitive, as long as you're already a Sky TV customer."
He compared Sky's Sim-only deal with that already offered by giffgaff, another service that piggybacks on O2's network and allows customers to alter their package depending on their usage.
"The key differences between the two are that Sky ties you in for 12 months, while giffgaff users can cancel anytime, and giffgaff also offers unlimited everything for £20 a month, beating Sky's top package, which delivers 5GB," said Mr Taylor-Gibson.
Sky's decision to let people roll over data and store it for three years was "both unique and compelling", he said, but it may have to offer bigger data bundles.
Aston Villa have signed Bristol City striker Jonathan Kodjia on a four-year deal, for an initial £11m that may rise to a Championship record fee of £15m.
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Broadcaster Sky is launching its own mobile phone service.