The island's 10 million people were spending their third night in the dark after the collapse on Friday of Cuba's largest power plant crippled the whole national grid.
The government said power is expected to be restored to most of the country by Monday evening.
But the president issued a warning there would no room for unrest in a country already battling sky-high inflation and shortages of food, medicine, fuel and water.
Dressed in a military uniform, Diaz-Canel said during a news conference that some Cubans had taken to the streets on Saturday evening in an attempt to "disturb public order."
Perpetrators will be prosecuted "with the severity that revolutionary laws provide," he said, adding that the protesters were acting "under the direction of the foreign operators of the Cuban counter-revolution."
Witnesses had reported that residents in several neighborhoods of Havana had taken to the streets on Sunday night to express their discontent.
There are people "making noise with pots and pans, shouting 'Let us have the power back on,'" a resident of the Santo Suarez neighborhood told AFP.
In July 2021, blackouts sparked an unprecedented outpouring of public anger.
Thousands of Cubans took to the streets shouting, "We are hungry" and "Freedom!" in a rare challenge to the government.
One person was killed and dozens were injured in the protests. According to the Mexico-based human rights organization Justicia 11J, 600 people detained during the unrest remain in prison.
- Storm Oscar weakens -
Cuba was still plunged in darkness on Sunday when Hurricane Oscar made landfall in its eastern part at 5:50 pm local time (2150 GMT) as a Category 1 storm.
It weakened into a tropical storm as it moved inland, the US National Hurricane Center said, whipping up waves up to 13 feet (four meters) high along the eastern coast.
Roofs and the walls of houses were damaged, and electricity poles and trees felled, state television reported.
President Diaz-Canel acknowledged in his address that the situation of the grid remained "complex," characterized by a high level of "instability."
Energy and Mining Minister Vicente de la O Levy told reporters Sunday that electricity would be restored for most Cubans by Monday night, adding that "the last customer may receive service by Tuesday."
The power grid failed in a chain reaction Friday due to the unexpected shutdown of the biggest of the island's eight decrepit coal-fired power plants, according to the head of electricity supply at the energy ministry, Lazaro Guerra.
- 'Cubans are tired' -
Power was briefly restored Sunday to a few hundred thousand inhabitants before the grid failed again, according to the national electric utility UNE.
Authorities have suspended classes and business activities until Wednesday, with only hospitals and essential services remaining operational.
Cuba is in the throes of its worst economic crisis since the collapse of key ally the Soviet Union in the early 1990s -- marked by soaring inflation and shortages of basic goods.
"Cubans are tired of so much... There's no life here, (people) can't take it anymore," lamented Serguei Castillo, a 68-year-old bricklayer.
"My fridge hasn't worked for three days now, and I'm afraid everything will be spoiled," 56-year-old worker Adismary Cuza.
President Diaz-Canel blamed the situation on Cuba's difficulties in acquiring fuel for its power plants, which he attributed to the tightening, during Donald Trump's presidency, of a six-decade-long US trade embargo.
The island is also feeling the aftershocks of the Covid-19 pandemic battering its critical tourism sector, and of economic mismanagement.
To bolster its grid, Cuba has leased seven floating power plants from Turkish companies and also added many small diesel-powered generators.
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