“We have won,” Mr. Putin declared to a huge throng of supporters right outside the Kremlin walls, a tear running down his cheek. “We have gained a clean victory!” He added, “We won! Glory to Russia.”

Mr. Putin has been Russia’s pre-eminent leader for 12 years, having served two terms as president from 2000 to 2008 before his current term as prime minister. But the opposition groups’ calls for more protests, starting with a rally on Monday night in Pushkin Square in central Moscow, threaten to undercut his promise of stability.