LVIV, Ukraine — To much of the world, Syria is a scene of unending tragedy, but to President Vladimir V. Putin of Russia it is a golden opportunity, a way out of the isolation he and Russia have endured since the West imposed sanctions over Ukraine — with the added bonus of wagging an “I told you so” finger at the White House.

His opening gambit to ingratiate himself with the West after a year of ostracism began with a singular gesture that Washington could hardly miss: dispatching a pronounced new flow of military hardware to Syria.

This week, Mr. Putin unleashed a diplomatic offensive, pushing to meet with President Obama, offering to hold military-to-military talks on Syria, and planning a big rollout for a Syrian peace plan when he speaks at the United Nations later this month.

The stakes for Mr. Putin are high — perhaps the highest in his career. The Kremlin has been on the defensive, diplomatically isolated after its adventures in Ukraine and battered economically by sanctions, low oil prices and a weak ruble that is cutting into living standards. Rapidly depleting the rainy day funds that have staved off financial disaster so far, Mr. Putin knows he needs to get back in the West’s good graces in a hurry, or at least change the conversation.