Speaking in an interview with Russian-language news agency RIA Novosti on Tuesday, Lavrov acknowledged the gravity of the problems between his country and the 28-member alliance.
He said: "Relations between Russia and NATO are going through the deepest crisis since the end of the Cold War … nevertheless, we consider it vital to keep the channels for political dialogue open."
Lavrov said Moscow and NATO should continue to fulfill their obligations according to the Russia-NATO 1997 Founding Act on Mutual Relations, Cooperation and Security.
Russia had no plans to close either its permanent mission to NATO or the NATO Information Office in Moscow as they perform tasks in line with the agreement, he added.
Moscow concerned by German leaders
Under Chancellor Angela Merkel, Berlin has taken the lead in trying to convince President Vladimir Putin to engage with the West during the crisis in Ukraine but her criticism of Moscow has sharpened in the last few weeks.
"We can't not express our concern over what our German colleagues are doing," he told a news conference after talks with Belgian Foreign Minister Didier Reynders.
"Germany has traditionally played a very constructive role regarding EU ties with Russia and the West as a whole with Russia. If Germany would decide to move to issuing orders, then Europe wouldn't win from that and neither would Germany."
Merkel spelled out her position on Tuesday to her conservative Christian Democrats (CDU), saying Russia's annexation of the Crimea peninsula in March was illegal and that the crisis in eastern Ukraine was threatening European order.
"We have had to witness a shifting of borders in Europe, the annexation of Crimea and the questioning of territorial integrity - a core part of Europe's post-war model. The apparent right of whoever is strongest trumped the strength of law," she said.
"Russia has breached and is still breaching international law...neighbouring states are suddenly no longer partner countries but spheres of influence," she said, underlining that only diplomacy could resolve the crisis.
German leaders have historically had close ties with Russia, but public opinion supports Merkel's increasingly hard stance against Moscow and the majority of Germans support EU sanctions against Russia, a poll for ZDF television showed last month.
After former Ukrainian president Viktor Yanukovych fled the country in the face of mass pro-European Union protests, Russia annexed Ukraine's Crimean peninsula, prompting the U.S. and many Western nations to impose sanctions against specific individuals in Russia and the Crimea.
The U.S., EU and several other nations levied the sanctions at the beginning in March, with successive rounds of punitive action being added due to Russia's alleged involvement with the conflict in eastern Ukraine.
Russia imposed an ban on food imports from states participating in the sanctions in response.
Russia has been accused of supporting the conflict in eastern Ukraine with soldiers and supplies, which Moscow has consistently denied.