Cypriot President Nicos Anastasiades has said a big bailout - which has provoked mass public anger - was needed to avoid a "disorderly bankruptcy".
The 10bn-euro ($13bn; £9bn) deal agreed by the EU was "a painful but controlled management of the crisis", he said.
Many Cypriots, shocked that the bailout imposes a levy on bank deposits of up to 10%, were seen queuing to withdraw cash.
The parliament is due to meet later on Sunday to vote on the measure.
Mr Anastasiades' Democratic Rally party - which has 20 seats in the 56-member assembly - needs support from other factions to ratify the bailout.
The deal - reached with eurozone partners and also the IMF in Brussels late on Friday - marks a radical departure from previous international aid packages.
In a statement on Saturday, President Anastasiades described the deal as choice between the "catastrophic scenario of disorderly bankruptcy or the scenario of a painful but controlled management of the crisis".
He said that thousands of small businesses would have gone bankrupt without the agreement.
The Cypriot leader, who was elected last month on a promise to tackle the country's debt crisis, is due to address the nation later on Sunday.
Nicholas Papadopoulos, who heads the parliamentary financial committee, said his initial reaction to the bailout deal was "shock".
"This decision is much worse than what we expected and contrary to what the government was assuring us, right up until last night (Friday)," Mr Papadopoulos told Reuters.
Meanwhile, opposition leader George Lillikas has urged his supporters to stage a protest rally on Tuesday, saying that the president "had betrayed the people's vote".
People in Cyprus with less than 100,000 euros in their accounts will have to pay a one-time tax of 6.75%, Eurozone officials said after agreeing the deal.
Those with greater sums will pay 9.9% in tax.
Depositors will be compensated with the equivalent amount in shares in their banks.
Reports suggest that depositors will be able to access all of their money except the amount set by the levy.
The levy itself will not take effect until Tuesday, following a public holiday, but action is being taken to control electronic money transfers over the weekend.
Co-operative banks, the only ones which were open in Cyprus on Saturday, closed after people started queuing to withdraw their money.
At one bank in the Limassol district, a frustrated man parked his bulldozer outside and threatened to break in.
Alan, a British expatriate saver in Cyprus, told BBC News: "This is robbery and we must get the EU to stop this.
"We retire and bring our savings to a bank in Cyprus and they can just take our money away without permission and then say we have shares in a bankrupt bank."
Maria Zembyla, from Nicosia, said the levy would make a "big dent" in her family's savings and "erode the investor confidence".
"Russians that currently keep the economy afloat will leave the country along with their money," she added.
According to Reuters news agency, almost half of the depositors in Cyprus are believed to be non-resident Russians.
Russians reacted angrily to the news of the levy on social media.
International lenders are gambling that the risk of a bigger banking crisis elsewhere in the eurozone has receded, the BBC's business editor Robert Peston writes.
While Cyprus may be one of the eurozone's tiniest economies - its third-smallest - there could be serious repercussions for other financially over-stretched economies, such as those of Spain and Italy.
The point of the levy is to warn lenders to banks that they should take care where they place their funds, and avoid banks that overstretch themselves - as Cypriot banks did, he adds.
Cyprus is the fifth country after Greece, the Republic of Ireland, Portugal and Spain to turn to the eurozone for financial help during the region's debt crisis.
The country has been in financial difficulties since the collapse of the Greek economy, where Cypriot banks had huge investments.