Sergey Lavrov's comments at the Munich Security Conference on Saturday came just hours after Pence promised to stand with Europe to rein in a resurgent Russia and "hold Russia accountable".
Lavrov, offering pragmatic ties with the United States, said: "I hope that [the world] will choose a democratic world order - a post-West one - in which each country is defined by its sovereignty."
The time when the West called the shots was over while NATO was a relic of the Cold War, he said.
In its place, Russia wanted a relationship with the US that is "pragmatic with mutual respect and acknowledgement of our common responsibility for global stability".
The two countries had never been in direct conflict, he said, and that they were close neighbours across the Bering Straits.
Russia has been impatiently waiting for US President Donald Trump to make good on his pledge to improve ties, which fell to a post-Cold War low as his predecessor, Barack Obama, declared sanctions over the Ukraine crisis and Russia's alleged meddling in the US presidential election.
On the campaign trail, Trump repeatedly praised Russian President Vladimir Putin and voiced his willingness to work with him in fighting "terrorism".
But in the face of growing heat over its links to Russia, Trump's administration appears to be backing off the warmer words used earlier for the country.
Exasperated and worried by Trump's calling into question long-standing foreign policy assumptions, European leaders have urged the US not to take transatlantic ties for granted.
On a European roadshow this week, Trump's lieutenants have sought to reassure jittery allies that the administration will hold fast to existing foreign policies, including maintaining sanctions on Russia over the Ukraine crisis.
Hours before Lavrov addressed the conference, Pence told the same forum that the US will stay loyal to its old friends.
"The United States is and will always be your greatest ally. Be assured that President Trump and our people are truly devoted to our transatlantic union," Pence said, adding that America strongly "supports NATO".
"Let no one doubt our commitment," he said.
Al Jazeera's Dominic Kane, reporting from Munich, said: "The warmth of words he used regarding NATO as a bedrock of security in Europe was warmly received here, by delegates, but not when he mentioned the need for NATO countries member states to boost the amount of their GDP spent on defence expenditure ... that causes some concern."
'New common ground'
The US would also not relent in pushing Russia to honour the Minsk ceasefire accords with Ukraine, Pence said.
"The United States will continue to hold Russia accountable, even as we search for new common ground, which as you know, President Trump believes can be found."
At NATO headquarters in Brussels on Thursday, James Mattis, defence secretary, said Russia must first "prove itself" and respect international law before there could be any improvement in relations strained by Russia's Ukraine intervention and annexation of Crimea.
Mattis also said the transatlantic bond was "as strong as I've ever seen it", and emphasised that America remained "rock solid" in its support of Article 5 - NATO's core "one for all, all for one" collective defence pledge.
Likewise, Rex Tillerson, US secretary of state, indicated he would take a tough line in his dealings with Russia.
Following his first meeting with Lavrov in Bonn on Thursday, Tillerson said the US would cooperate with Russia but only when doing so "will benefit the American people".
German Chancellor Angela Merkel, who has been a key player in mediating talks between Ukraine and Russia, said it was "regrettable" that Europe had not managed to reach a stable relationship with Russia over the last 25 years.
"I will not give up on finding a way for better relations with Russia despite our different views on many questions," she said, championing international cooperation rather than a policy of isolationism.
"In a year in which we see unimaginable challenges, we can either work together or retreat to our individual roles. I hope that we will find a common position," she said.