The agency saw off challenges from McCann Erickson, BBDO and TBWA to win the account.
It will handle through-the-line and event communication, raising awareness of Turkcell's Super League sponsorship, and its sponsorship of the national football and basketball teams.
Meral Akyel, the managing director of Leo Burnett Turkey, said: "This was the biggest pitch of the year, the Holy Grail, and we're thrilled that we beat some of the big players in the industry."
With almost 53 million customers, Turkcell has a market share of around 60 per cent. Its nearest rival, Vodafone, has 24 per cent.