The single currency had come under pressure in recent weeks on worries Marine Le Pen's National Rally would storm to victory and push through big-spending policies that could put the country on a collision course with the European Union.
However, while polls suggested the party was on course to win a majority of the 577-seat National Assembly, it was unclear if it could garner enough to take full control.
President Emmanuel Macron called for a "broad" alliance against the far-right in the second round, which will see run-off votes where there was no outright winner in the first round.
The euro climbed to as high as $1.0762 Monday.
The yen meanwhile held its own against the dollar after the personal consumption expenditures (PCE) index -- the Federal Reserve's preferred gauge of inflation -- eased further in May.
The reading provided fresh optimism that the Fed will cut interest rates this year, while traders at the same time bet on the Bank of Japan lifting borrowing costs at its next meeting.
Traders are now looking ahead to the release of key non-farm payrolls figures at the end of the week.
"The door to a September cut is wide open, but three payrolls and three (consumer price indexes) ahead for the September meeting still need to go by without giving (decision-makers) too much to be nervous about," said Taylor Nugent at National Australia Bank.
"That's far from guaranteed."
Traders are keeping a close eye on developments in forex markets after Japanese officials warned they were ready to step in to support the yen if it continues to weaken. The unit is hovering just below 161 per dollar, having hit a 38-year low last week.
Equity markets were mixed after a tepid lead Friday from Wall Street, where all three main indexes struggled to hold early gains and ended slightly lower.
Still, they are up handsomely in 2024 so far, with the S&P 500 piling on more than 14 percent.
Tokyo edged up in the morning session as the BoJ's key Tankan survey of Japan's largest manufacturers showed business confidence rose for the first time in two quarters.
There were also gains in Seoul, Singapore, Taipei, Manila, Jakarta and Wellington, though Shanghai and Sydney edged lower.
Hong Kong was closed for a holiday.
There was little major reaction to data showing Chinese factory activity shrank for a second straight month in June, highlighting ongoing weakness in the world's number two economy ahead of a key political gathering this month.
- Key figures around 0230 GMT -
Tokyo - Nikkei 225: UP 0.3 percent at 39,687.40 (break)
Shanghai - Composite: DOWN 0.1 percent at 2,964.40
Hong Kong - Hang Seng Index: Closed for holiday
Euro/dollar: UP at $1.0756 from $1.0715 on Friday
Dollar/yen: DOWN at 160.87 yen from 160.92 yen
Pound/dollar: UP at $1.2660 from $1.2644
Euro/pound: UP at 84.91 pence from 84.71 pence
West Texas Intermediate: UP 0.4 percent at $81.90 per barrel
Brent North Sea Crude: UP 0.4 percent at $85.35 per barrel
New York - Dow: DOWN 0.1 percent at 39,118.86 (close)
London - FTSE 100: DOWN 0.2 percent at 8,164.012 (close)
dan/dhw
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