Chanturia began his career on FC Saburtalo, and was loaned to Barcelona in June 2009. He was filled into Barça's Juvenil B squad, and scored 11 goals to help the Catalans to the league title.
In August 2010, after been linked with Chelsea, Chanturia signed with Vitesse. In April 2011 he signed a three-year deal with the club.
Chanturia made his competitive debut for Vitesse on the opening day of the 2011–12 Eredivisie season, as a starter in their 0–0 away draw against ADO Den Haag on 7 August. He scored his first goal for the Arnhem side in his second official match, against VVV-Venlo in a 4-0 win.
In December 2012, Chanturia was loaned to Alania Vladikavkaz until the end of the season. He returned to Arnhem club at the end of the season, and was reintagrated to first-team squad for the 2013–14 season.
On 19 February 2014, CFR Cluj have confirmed the signing of Vitesse midfielder Giorgi Chanturia for an undisclosed fee.
He made his first international appearance for Georgia in a friendly match against Liechtenstein on 5 March 2014. He scored only 25 minutes into his debut.