"We'll need to persuade them (the protesters) not to violate law and order, not to violate others' rights. But if they do, then the law-enforcement bodies will need to restore law and order."
Official results gave Sarkisian, a close ally of Robert Kocharian, the outgoing president, just under 53 per cent of the vote, compared with 21.5 per cent for his nearest rival, Levon Ter-Petrosian, a former president.
Ter-Petrosian's supporters say the election was marred by ballot stuffing and intimidation.
Gaghik Jahangirian, Armenia's deputy proscutor-general, addressed the demonstrators and told them to keep up their protests.
It is extremely unusual for high-ranking officials to express sympathy with the opposition.
Jahangirian said: "I've seen many elections in this country but I've never seen the level of violations and violence as this time."
Western election monitors said the ballot was broadly in line with the country's international commitments but that further improvements were necessary.
Kocharian and Sarkisian are both natives of Nagorno-Karabakh, a region over which Armenia and neighbouring Azerbaijan fought a war in the 1990s.
Some analysts say the still unresolved conflict could flare again into violence.
During the war, Turkey closed its border with Armenia and froze diplomatic relations in solidarity with Turkic-speaking Azerbaijan.
Relations have still not been restored, and are also complicated by the massacres of Armenians by Ottoman Turks during World War I.
Turkey strongly denies Armenian declarations that the killings were genocide.
Sarkisian said: "I've declared many times that Armenia is ready to establish diplomatic relations with Turkey without any pre-conditions."
Abdullah Gul, Turkey's president, congratulated Sarkisian this week on his election win and said he hoped for an improvement in relations between their countries.