MOSCOW—Russian President Vladimir Putin paid a surprise visit to a military air base in Syria, in an effort to showcase his country’s rising clout in the Middle East months ahead of a presidential election.

Mr. Putin landed Monday at Hmeimim air base, a military outpost in Syria’s Latakia province that has been the main launching point for Russian warplanes backing the forces of Syrian President Bashar al-Assad. In remarks to Russian troops, he claimed victory, saying Russian forces and their allies had “defeated the most...