Currency traders sold off the euro over fears about the uncertain future for Greece in the eurozone following the results of the vote, which showed more than 61 percent of the electorate voting no while 39 percent supported the yes side.
The euro fell one percent to $1.099 from $1.111 against the dollar, while it slid to as little as 1.036 francs from 1.045 francs as soon as polls pointed to a “no” victory.
The euro rebounded on Monday, however, after Greek finance minister Yanis Varoufakis announced he was quitting, stating that fresh negotiations between the Greek government and creditors would go better without his presence at the table.
"Soon after the announcement of the referendum results, I was made aware of a certain preference by some Eurogroup participants, and assorted ‘partners', for my… ‘absence' from its meetings," he wrote on his blog.
Varoufakis, who had earlier accused creditors of being "terrorists", said stepping down was "an idea that the Prime Minister judged to be potentially helpful to him in reaching an agreement. For this reason I am leaving the Ministry of Finance today".
Greek Prime MinisterAlexis Tsipras, leader of the left-wing Syriza party elected in January to oppose austerity measures that have shrunk the Greece economy, said the referendum result did not mean that voters have given the government a mandate against Europe.
They have instead given it a mandate for a “sustainable solution,” he said.
Greek television reported that Tsipiras had started on Sunday night calling European leaders, including French President Hollande who was to meet with German Chancellor Angela Merkel about the Greek crisis.
But it was unclear what kind of cooperation Greece could expect from other European countries.
The head of Germany's savings bank association told Reuters that Greece should leave the eurozone.
And the head of the German exporters association said he didn't see how Greece could remain in the single-currency bloc.
Sigmar Gabriel, Germany's economy minister, told the Tagesspiegel newspaper that the referendum result makes it difficult to imagine talks on a new bailout programme with Greece.
Some financial analysts said the yes vote would force Greece out of the eurozone, something that 75 percent of Greek voters oppose, according to one poll.
The uncertainty has driven up the value of the Swiss franc, seen as a safe haven in times of crisis.
Johann Schneider-Ammann, Swiss minister of economic affairs, said he expected the franc would be impacted by the result of the Greek referendum.
"The pressure on the franc is maybe going to accentuate," Schneider-Ammann, currently on a five-day trip to the US, told Le Temps newspaper.
"The critical situation that the eurozone finds itself in shows us the importance of diversifying our export markets."
Schneider-Ammann added that the Swiss National Bank (SNB) was independent and it was up to it "to act if it sees fit".
The central bank has already been intervening in foreign exchange markets to prevent the currency from rising further.
Traders last Monday drove the euro down to 1.0314 francs before the SNB intervened by buying euros.
The euro briefly head above 1.05 francs before falling back.
SNB Chairman Thomas Jordan said the central bank was prepared for a Greek euro exit but he did not give details about the strategy.
The bank in January stopped defending a 1.20 franc-euro rate after supporting this for more than three years, saying it was unsustainable, while at the same time claiming the franc was significantly overvalued.
A strong franc hurts Swiss exporters because it makes their products and services more expensive to customers in the eurozone, Switzerland's largest trading partner.
Swiss media commentators were skeptical about the chances of Tsipras reaching a deal with his European partners and creditors.
Commentator Peter A. Fischer, in an online column for the Nueue Zürcher Zeitung, said a “Grexit” is now a likely outcome from Sunday's vote.
“A Greek exit from the monetary union cannot be enforced but it is the logical consequence of the people saying no,” Fischer said.
“The rude awakening will not be long in coming,” Stephan Israel, Tages Anzeiger's EU correspondent said.
He said the prospects were bleak in Greece, where banks have been closed for a week and capital controls introduced following the collapse of talks with creditors and the announcement of the referendum.
Tsipras's hopes of rapidly reaching a new agreement with loans from creditors after the no vote will come to nothing, Israel said.
“Not the least because the do not countries have an increasingly critical public, which it must take into consideration,” he said.
“Tsipras and his associates have destroyed any basis for trust.”