Iraq joined Russia, Iran and Syria in a new agreement to strengthen cooperation against extremist group Islamic State, extending the Kremlin’s reach in the Middle East as it rivals Washington for influence.

U.S. and Russian officials held talks Sunday on the sidelines of a United Nations summit in New York to try to forge a common approach to fighting Islamic State, a day before President Barack Obama and Russian President Vladimir Putin were to hold their first formal meeting in more than two years at the U.N. The two had...