Foreign Minister Sergei Lavrov made the comments in a phone call with his American counterpart John Kerry.
Western sanctions, coupled with falling global oil prices, have hurt Russia's economy.
The EU and US have imposed a string of financial penalties on Moscow following Russia's annexation of Crimea in March.
The new US legislation, aimed mainly at Russia's defence industries, gives President Barack Obama the power to impose further sanctions if he chooses - though no action has yet been taken as a result of the bill.
It would also allow Mr Obama to provide lethal and non-lethal military assistance to Ukraine.
On Thursday the EU introduced measures which outlaw European investment in Crimea, the Black Sea peninsula annexed by Russia.
During a marathon end-of-year press conference on Thursday, Mr Putin accused Washington and the West of trying to isolate Moscow.
He said the US was trying to tame the Russian "bear", adding that sanctions were to blame for up to 30% of the economic woes which led to the dramatic slide of the rouble on Tuesday.
Washington is calling on Mr Putin to pull out of the Crimea and stop supporting pro-Russian rebels fighting the Ukrainian government.
Mr Obama has said the US is ready to "roll back" sanctions if Russia takes the "necessary steps" to defuse the current standoff.
Talks are due to start on Sunday involving Ukrainian officials and pro-Russian separatists.
The negotiations are aimed at implementing a ceasefire deal that was signed three months ago but which has failed to halt fighting.
Ukraine's National Security and Defence Council said on Friday that five soldiers were killed and seven injured during clashes in the east of the country.
Russia has warned that a new US law authorising fresh sanctions against Moscow could damage relations between the two countries "for a long time".