Thousands of anti-government protesters have been rallying the streets of Yerevan since April 13, and Sunday's rally attracted some 50,000 demonstrators.
"[Opposition leader] Nikol Pashinian was right", he said in a written address to the nation.
"This is not in my nature", he added.
On Sunday, protest leader Nikol Pashinyan was detained, but was released earlier on Monday.
Sargsyan made the announcement in a statement on his website, saying that "the movement of the streets is against my office", so he chose to leave office. "You won, proud citizen of the Republic of Armenia".
Mr Pashinyan congratulated the people on their "victory" following the resignation.
Former prime minister Karen Karapetyan has taken over as acting PM.
Large anti-government protests began almost two weeks ago, after the outgoing president, Sargsyan, was nominated as the ruling party's candidate for prime minister. He also held the office of prime minister from 2007 to 2008.
Touching on a possible introduction of a special regime at the Russian military base in Gyumri, the spokesman said this was a matter of the Russian Defense Ministry.
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"The emotional catalyst was the decision of Serzh Sarkisian to stay in power..."
I am addressing you all for the last time.
"This has never happened before".
Although Sargsyan is gone, Shirinyan said the hard work of rebuilding Armenia's checks on political power and securing its democratic foundation remains.
"Who will be the prime minister?" The demonstrations had reiterated ir willingness to continue with concentrations until resignation of Sargsyan, an exmilitary of 63 years, who blames for not having reduced poverty or corruption, while magnates continue to have an advantage over Economy of country. "We will have to live through all of this".
"I believe that the fact of negotiations itself, despite all the formats aimed at hampering the talks, starting from holding them in front of media, at Marriott Hotel, shows the Prime Minister's and government's willingness to launch a dialogue", he said.
It was not immediately clear whether Sarkisian's exit will be followed by fresh parliamentary elections. His chosen successor as president, Armen Sarkissian - no relation - was sworn in a week previously after also being elected by parliament.
The Kremlin follows "closely situation in Armenia", an "extremely important country" for Russian Federation and "close ally", said in Moscow spokesman of President Vladimir Putin, Dimitri Peskov.
Peskov dismissed a question on whether Russian Federation would interfere in the crisis, which he called "exclusively an internal affair", as "absolutely inappropriate".