The Indiana Republican, a member of the Senate Intelligence Committee, added that any new sanctions should be imposed immediately, saying it would be a mistake to wait to act until after Ukraine's scheduled national elections on May 25.
Coats said sanctions should target Russia's economy, saying the United States and its allies are "playing right into his hands by not imposing sanctions that really hit [Putin] where it hurts."
President Obama in March froze the U.S. assets of several Russian officials, including top advisers to Putin, for their support of Crimea's contested vote to secede from Ukraine.
But the senator accused the Obama administration for being too soft on the Russian president, saying sanctions so far have been "too little, too late."
"They should have been imposed much earlier and should go right to the Russia pocketbook," he said. "That is ultimately what is going to turn the tail here."
"Clearly Putin has the upper hand here. He sees weakness from the West, weakness from leadership here in the United States."
Coats joined with a group of Republican senators who have introduced legislation that would provide a "strategic U.S. response" to Russian actions in Europe.
Meanwhile, Canadian Prime Minister Stephen Harper on Sunday said his government is imposing economic sanctions on 16 Russian "entities" in response to its "illegal occupation of Ukraine and provocative military activity."
The Harper government already had imposed political and economic sanctions against senior officials and some institutions in Russia and Ukraine.