Russian U.N. representative Vassily Nebenzia pushed back and said much of what the vice president said came from Guaido.
But it is unlikely Washington will get the support needed to carry out something so huge in either the Security Council or the General Assembly.
Borrowing Trump's campaign slogan "Make America Great Again", Nebenzia said: "If you want to make America great again, and we're all sincerely interested in seeing that, stop interfering in the affairs of other states".
Moncada also stressed that the Venezuelan diplomats are restricted in terms of movement in the U.S. - they are only allowed to go on trips of up to 25 miles away from NY.
Although Maduro and his socialist ruling party enjoy support from many poor and working-class Venezuelans, his government has been facing its toughest challenge to date from Guaidó, who has highlighted the country's plummeting economy, as well as the exodus of Venezuelans, in calling for Maduro's ouster.
"There is a clear move here to undermine our rights", Moncada said, "and if they can undermine our rights, they can undermine the rights of all members of this organization, so we must categorically reject this".
United Nations humanitarian chief Mark Lowcock said the crisis has worsened and that 25 percent of the population need humanitarian aid.
Asked what made the US think it has support to give Guaido UN recognition, Pence said, "I think momentum is on the side of freedom - momentum is on the side of the suffering people of Venezuela".
He briefed the council on a recent United Nations report on the situation that estimates about a quarter of Venezuelans are in need of humanitarian assistance, and painted a dire picture of millions of people lacking food and basic services.
Following Pence's speech, Russia's United Nations envoy Vassily Nebenzia said that the U.S. had "artificially provoked a crisis" in Venezuela in order to overthrow "a legitimately elected leader and replace him with their own pawn". The attempted passage of the convoys resulted in violence with photos of a burning aid truck becoming a source of worldwide outrage and a highly-publicized symbol of Maduro's "dictatorship".
"In Venezuela, there is a need to separate political and humanitarian objectives".
"That's the nature of the regime that we're dealing with".