In a message carried by the official APS news agency on Monday, the 82-year-old also said the elections would follow a national conference on political and constitutional reform to be carried out by the end of 2019.
"There will be no presidential election on April 18," Bouteflika said in reference to the scheduled date of the vote, adding he was responding to a "pressing demand that you have been numerous to make".
The dramatic developments followed weeks of mass demonstrations against Bouteflika's plan to extend his 20-year rule. The unprecedented citizens' revolt drew millions to the streets of cities across the country.
Celebrations popped up instead of protests on the streets of the capital, Algiers, at Monday's news. Car horns rang out while people waved flags, jumped up and down, and sang the national anthem.
Others were more cautious, calling their long-time leader's pledge to step aside just a first step. Bouteflika did not give a date or timeline for the delayed election.
Critics said they fear the moves could pave the way for the president to install a hand-picked successor. Others saw his decision to postpone the election indefinitely as a threat to democracy in Algeria.
|WATCH: Algeria protests – Partial strike under way in the capital (01:44)|
"Even if this is a beautiful victory for the Algerian people and the gesture was there, I do not believe that the entire regime and its system is going to collapse," Dalia Ghanem Yazbeck, a resident scholar at the Carnegie Middle East Center, told Al Jazeera.
"This is a regime that is composed of different strata and circles of power. You have the [ruling party] FLN apparatchik, you have the bureaucracy, political and military leadership and you have business tycoons," she added.
The ailing leader, who has been confined to a wheelchair since suffering a stroke in 2013, said a government reshuffle would also take place.
According to APS, Prime Minister Ahmed Ouyahia resigned on Monday and was replaced by Interior Minister Noureddine Bedoui. Ramtane Lamamra was named deputy prime minister, a position that did not exist before.
'All eyes on the army'
Amel Boubekeur, a research fellow at the Paris-based School for Advanced Studies in the Social Sciences, said Monday's announcement was only the beginning.
"All eyes are on the army now. Is the army going to let new protests to happen next Friday?" Boubekeur told Al Jazeera.
"With Bouteflika aside, the army is going to have its say as to what kind of position they will accept.
"Now it seems the regime of Bouteflika is done, but [the question is] are we going to get back the state civilly without … any pressure on all these people who have been using the state for their interest for so long?"
A veteran of the country's war of independence against colonial France, Bouteflika has seen his popularity wane in recent years as a result of his deteriorating health.
On Monday, Algerian state television aired the first images of Bouteflika since the protests started. He appeared weak and moved with slow gestures. No sound accompanied the images.
Massive protests began on February 22 to denounce Bouteflika's plans to extend his rule in the upcoming polls.
On March 3, after his campaign manager officially registered Bouteflika's candidacy, the president tried to appease protesters by offering to hold a national dialogue conference, change the constitution and hold a vote within a year of his reelection in which he promised not to take part.
The promises, however, failed to quell public anger, galvanising discontent among different sectors, particularly students and other young people.
'System is over'
Some long-time allies of Bouteflika, including members of the FLN party, expressed support for the protesters, revealing cracks within a ruling elite long seen as invincible.
|WATCH: Algeria's protests explained (07:25)|
In the clearest indication yet that the generals sympathise with protesters, the chief of staff said on Sunday the military and the people had a united vision of the future, state TV reported. Lieutenant General Gaid Salah did not mention the unrest.
"Bouteflika's system is over," said a commentator on Ennahar, which is close to the president's inner circle.
In response to Monday's announcement, Soufiane Djilali, leader of the opposition Jil Jadid party, said Bouteflika could not stay in power until late 2019.
"The first round has been won. The mobilisation on the streets must not stop. There can be no question of the current presidency continuing until the end of 2019! Zero confidence," Djilali wrote on Twitter.
|WATCH: Inside Story – How will Algeria's army handle protests against president Bouteflika? (24:50)|
SOURCE: Al Jazeera News