Mr Lavrov also said allegations of Russian cyber attacks were fabricated, amid allegations from the United States that Moscow tried to influence November's US election.
The minister told a press conference that US intelligence agencies who had tried to prove Mr Trump had links to Russia have drawn a blank and should be fired.
Mr Trump, who will be sworn in on Friday, has dismissed the dossier, saying it was "made up" and based on "phony facts".
Moscow said the claims were "fake" and had been whipped up to damage Russian ties with Washington.
Mr Lavrov said Moscow is hopeful of better relations with the US based on respect for mutual interests once Mr Trump is inaugurated, in contrast with the "messianic" approach of Mr Obama.
He said Russia expects to have a dialogue with Mr Trump on nuclear weapons, which could include hypersonic weapons, a US missile shield in Europe, space weapons and nuclear testing.
Mr Lavrov said he would like to see Mr Trump's team made up of pragmatic people "who will not engage in moralising and will try to understand the interests of their partners just as they clearly uphold their own interests".
SKY News.
