The 37-year-old, who lost the Champions League final with Inter to Manchester City in Istanbul earlier this month, will receive 4.2 million euros ($4.6 million) per season, the club said.
Dzeko scored 14 goals in all competitions for Inter last season, but was out of contract with the side he joined from Serie A rivals Roma in 2021.
He previously played for Manchester City and Wolfsburg, winning Premier League and Bundesliga titles.
Dzeko, who has netted more than 300 goals in club football, is also the all-time top-scorer for Bosnia and Herzegovina with 64 international goals in 129 games.
Fenerbahce, runner-up to their great rivals Galatasaray last season in the Super Lig, have not won the Turkish top-flight title since 2014.
The club won the Turkish Cup in early June, their first trophy in nine years.
Dzeko will fill the gap left by Ecuadorian Enner Valencia, top scorer in the Super Lig with 29 goals last season, who has left for Internacional in Brazil.
