It fell as low as $1.0560, before recovering a little. But many traders expect it may soon be worth the same as a dollar.
The ECB began its latest round of quantitative easing (QE) on Monday.
It will buy bonds worth â‚¬1.14tn over the next 18 months, flooding the market with euros.
Traders have reacted to the ECB's latest round of QE by selling euros and buying other currencies such as US dollars.
The US currency is appealing because the Federal Reserve looks to have completed its bond-buying programme.
The euro started its slide against the dollar in July last year as traders reacted to the divergence in policy between the ECB and the Fed.
The value of the euro has fallen 22.4% since 1 July, when a euro was worth $1.37.
An upbeat US jobs survey released on Friday provided an additional boost to the dollar.
"This opened up speculation again that the US will raise interest rates in June," says Jane Foley, senior currency strategist at Rabobank. This would attract foreign capital and boost the dollar.
In the eurozone Greece's economic woes continue to put downward pressure on the euro, bringing it closer to dollar parity.
The eurozone's growing current account surplus is encouraging Europeans to invest abroad causing the euro to weaken further, according to Deutsche Bank.
It's what you expect when the economic performance of two currency blocs diverge in the way that the eurozone and the US have.
The stronger growth in the US means higher interest rates, because the Federal Reserve will raise its own rates, perhaps later this year, and also because there is more demand for cash to fund investment.
In Europe by contrast, the quantitative easing that is finally underway is driving down borrowing costs in the financial markets even further.
The weaker euro is also what you want, at least if you are at the eurozone end of this exchange rate.
Part of Europe's problem is weak demand for goods and services at home and the cheaper currency will make it a little easier to compensate for that by selling more abroad.
"The momentum is certainly building and there's a lot of talk of parity," says Ms Foley.
"We now see euro-dollar moving down to $1.00 by year-end, $0.90 by 2016 and down to a trough of $0.85 by 2017," said Deutsche Bank in a report published on Tuesday.
While further devaluation of the euro should give businesses in the eurozone a boost, the pace of change may pose challenges for companies that need to plan ahead.

A spokesman for the US authorities said the operation had targeted high-level members of the cartel.
The spokesman said the operation had dealt a strong blow to the criminal organisation.
It comes weeks after the leader of the Sinaloa cartel, Joaquin Guzman was arrested after six months on the run.
Inside the feared Sinaloa cartel
'El Chapo' Guzman and the Sinaloa cartel
The spokesman, for the US Immigration and Customs Enforcement (ICE) agency said they had helped Mexican police target a Sinaloa cartel cell operating around the Mexican city of Sonoyta, along the border with the US state of Arizona.
He said it was responsible for importing millions of dollars of illegal drugs, including cannabis, heroin, cocaine and methamphetamine into the US from Mexico during its existence.
The operation was conducted in total secrecy with police working in both Lukeville and Sonoyta, bordering cities that are on the route to the Puerto Penasco, the popular beach destination many Americans know as Rocky Point.
Mexican police officers had been allowed into the US to keep them safe during the operations the US immigration authorities had said.
The arrests are the latest blow to the Sinaloa cartel, following the recapture of Guzman, also known as "El Chapo" (Shorty).
He was arrested a few weeks ago after six months on the run, following his second escape from a high-security Mexican prison.
According to Forbes Magazine, Guzman is the world's most powerful drug trafficker.
The cartel is responsible for an estimated 25% of all illegal drugs that enter the U.S. via Mexico.
Drug enforcement experts estimate, conservatively, that the cartel's annual revenues may exceed $3 billion.
This February the city of Chicago branded him the first "Public Enemy No. 1" since Al Capone.
The euro has fallen to its lowest level against the US dollar in 12 years after the European Central Bank (ECB) began its government bond buying programme.
[NEXT_CONCEPT]
Mexican police working with US agencies have arrested 24 members of the powerful Sinaloa drug cartel in a joint border operation.