By Andrew Cawthorne and Girish Gupta

CARACAS (Reuters) - Ecstatic opposition leaders vowed on Monday to use their new majority in Venezuela's legislature to free jailed opponents of the Socialist government but also said they would not move to dismantle popular welfare policies.

The opposition Democratic Unity coalition won more than twice the number of National Assembly seats as the Socialists in elections on Sunday that punished President Nicolas Maduro's government for the country's deep economic and social crisis.

It was the first time in 16 years the "Chavismo" movement, named for former socialist President Hugo Chavez, lost its majority in the 167-member assembly, and gives the opposition a platform to further erode Maduro's power in the OPEC nation.

The 53-year-old president, who was handpicked by Chavez but lacks his charisma and political guile, quickly accepted defeat in a speech to the nation that calmed fears of violence.

Aware that victory owed more to public discontent with Maduro than love for the opposition, coalition head Jesus Torrealba urged Venezuelans to bury their differences.

"We have been divided for years and the country has won nothing with this historic mistake ... The Democratic Unity is not here to mistreat anyone," Torrealba, who was mocked by Maduro as an "evil Shrek" during the campaign, told supporters in a victory speech in the early hours of Monday.

Reiterating that an Amnesty Law will be the opposition's priority when the new assembly begins work on Jan. 5, Torrealba promised to return the rights of "those who have been unjustly persecuted, jailed, blocked from politics or exiled".

Venezuela's best-known jailed politician is Leopoldo Lopez, sentenced to nearly 14 years on charges of promoting political violence in 2014 that killed 43 people. But the opposition has a list of what it says are more than 70 other political prisoners.

Torrealba also reassured despondent government supporters the coalition would not try to dismantle welfare programs that were wildly popular during Chavez's 1999-2013 rule and which Maduro repeatedly warned they want to end.

With 99 seats to the Socialists' 46 in counting so far - and results not yet in for the remaining 22 seats - the opposition looks certain to reach a three-fifths majority, meaning they could in theory have ministers fired after a censure vote.

With two-thirds, they could try and shake up institutions like the courts widely viewed as pro-government.

"The sheer scale of its victory could potentially give the opposition real teeth as it tries to alter the course of government policy under Mr Maduro," said Fiona Mackie, Latin America analyst at The Economist Intelligence Unit.

AMNESTY LAW?

Even with just a simple majority, the opposition can exercise control over the budget, begin investigations that could embarrass the government, and pass the amnesty law.

Torrealba has also said the assembly will open an investigation into the arrest of two relatives of Maduro, cousins of his wife, caught in a sting in Haiti and indicted in a New York court on charges of cocaine smuggling.

The United States, which has had an acrimonious relationship with Venezuela under both Chavez and Maduro, has long accused the Socialists of complicity in the drug trade, as well as human rights abuses.

The government calls that lies and frequently recalls U.S. support for a short-lived 2002 coup against Chavez.

A former bus driver and foreign minister who narrowly won election in 2013 after Chavez died from cancer, Maduro may face a backlash in the ruling party and from grassroots supporters who think he has betrayed his predecessor's legacy.

Though his term ends in 2019, hardline opposition leaders want to oust him in a recall referendum next year. They would require nearly 4 million signatures to force the recall vote.

"I can't see this government finishing its term because it is too weak," said opposition leader Henry Ramos, touted as a possible leader for the new assembly. "Internal frictions are beginning. They're blaming each other for this huge defeat."

Maduro, whose government has replaced Cuba as the most vocal adversary of the United States in Latin America, blamed an "economic war" waged by business leaders and other opponents out to sabotage the economy and bring him down.

"In Venezuela, a counter-revolution won, not the opposition," he said in a speech in the early hours.

"The economic war has triumphed for now."

Venezuelans have not in large bought that argument, though, blaming him for the world's highest inflation, shortages from milk to medicines, and a devalued currency that trades on the black market at nearly 150 times its strongest official rate.

Maduro's persistence with policies like complex currency controls have contributed to Venezuela's economic distortions but, unlike Chavez, he also had the misfortune to see a plunge in the price of Venezuela's only significant export, oil.

"This is Nicolas Maduro's defeat, not Chavez's," said Humberto Lopez, 57, a diehard Chavista well-known to Venezuelans for dressing like Argentine revolutionary Ernesto "Che" Guevara at government rallies. "I'm not hugely surprised."

Underlining the unprecedented mood in Venezuela, videos circulating online seemed to show five prominent socialist politicians - including Chavez's brother Adan - being booed at voting centers on Sunday, with crowds yelling "the government will fall!" or "thief!".

Foreign markets reacted positively, with dollar bonds rising strongly on hopes of business-friendly change.

The government's defeat was another disappointment for Latin America's bloc of left-wing governments following last month's swing to the center-right in Argentina's presidential election.

But words of consolation came from the Venezuelan government's closest ally, Communist-run Cuba.

"I'm sure new victories for the Bolivarian and Chavista Revolution will come under your leadership," President Raul Castro wrote to Maduro, referring to Venezuelan independence hero Simon Bolivar as well as his late friend Chavez.

(Additional reporting by Deisy Buitrago and Corina Pons in Caracas, Daniel Trotta in Havana, and Sujata Rao in London; Editing by Kieran Murray)