Entry visas for Algerians were introduced in 1994 when the land border between the two countries was closed. It followed an attack on Spanish tourists in Marrakesh by Islamist dissidents from Algeria.
The abolition of visas became effective on Friday and was announced hours after King Muhammad's address to the nation on the fifth anniversary of his rule.
There was no immediate reaction from the Algerian government. It was not clear if Algiers would lift visa requirements for Moroccans travelling to Algeria.
Morocco and Algeria have had prickly relations since their independence from long-time colonial ruler France, in 1956 and 1962 respectively.
Efforts to normalise ties have been hampered by the Western Sahara dispute. Morocco seized the vast desert territory in 1975 after Spain relinquished it.
Algeria backed the Polisario Front independence movement, which fought a guerrilla war with Rabat until a UN-brokered ceasefire in 1991.
"Rabat chooses pragmatism"
Le Quotidien d'Oran,
French language daily
The two countries have built up diplomatic exchanges in the past few years and bilateral ministerial visits have become commonplace.
A brief official statement announcing the abolition of visas on Saturday said it "reflected the sincerity and determination of the Kingdom of Morocco to place its relations with neighbouring Algeria within a new dynamic".
Algerian independent media said the decision would boost the Moroccan economy, in particular its buoyant tourism sector.
Top selling French language daily Le Quotidien d'Oran said on its front page: "Rabat chooses pragmatism. It's good news for the Moroccan economy but also for the human relations between the two countries."