"We are confident that the new administration does not want to repeat the errors of the outgoing one, which deliberately destroyed US-Russian relations," Lavrov was quoted as saying in an interview with Italian daily Corriere della Sera.
Trump's election victory and the positive noises he has made towards Moscow have been greeted with trepidation in Ukraine and former communist states that are now part of the US-led NATO alliance.
During the campaign, Trump praised President Vladimir Putin and said he would seek to improve relations with Moscow while casting doubt on Washington's commitments to its NATO allies.
Lavrov told Corriere: "Naturally we positively welcomed the willingness for cooperation between our two countries shown by Trump during the election campaign.
"On our part we are always available for a honest, pragmatic dialogue with Washington on all bilateral and global questions..."
He added: "We hope that the new president's fledgling foreign policy team will take concrete steps in this direction and that the cooperation will be constructive."
Lavrov defended Russia's build-up of forces in areas close to ex-Soviet NATO states as a response to the Western allies "political-military pressure on our country" which had obliged Moscow to "take appropriate measures for our defence and national security."
Lavrov was in Rome to attend a conference on Mediterranean security issues, during which he was due to meet with his US counterpart, John Kerry.