Following more than three hours of debate, German lawmakers voted 439-119 in favor of opening detailed discussions on the package. There were 40 abstentions.
The German Parliament's vote capped a week in which the proposed bailout agreed Monday by the 19 eurozone leaders, including Merkel and Greek Prime Minister Alexis Tsipras, has cleared a string of hurdles.
The developments have raised expectations that Greece will secure a financial lifeline to allow the country to get back toward some sort of economic normality following a crisis that has seen banks shuttered for nearly three weeks and withdrawals at ATMs limited to a paltry 60 euros a day.
Germany is one of the few eurozone countries whose parliaments had to approve the step. Earlier Friday, Austrian lawmakers also cleared the way for the talks.
Though the broad outlines of the bailout were agreed Monday, specific terms will now be thrashed out between Greece and its European creditors. The process is expected to last around four weeks and to lead to Greece getting around 85 billion euros ($93 billion) to help it pay off upcoming debts.
Germany has been the largest single contributor to Greece's bailouts and has taken a hard line, insisting on stringent spending cuts, tax hikes and wide-ranging economic reforms in return.
"The principle ... of responsibility and solidarity that has guided us since the beginning of the European debt crisis marks the entire result from Monday," Merkel told the special session of Parliament.
The alternative to an agreement, she added, "would not be a time-out from the euro that would be orderly ... but predictable chaos."
Merkel will have to return to Parliament to seek approval for the final deal when the negotiations are concluded.
"I know that many have doubts and concerns about whether this road will be successful, about whether Greece will have the strength to take it in the long term, and no one can brush aside these concerns," she said. "But I am firmly convinced of one thing: we would be grossly negligent, even irresponsible, if we did not at least try this road."
Merkel's finance minister, Wolfgang Schaeuble, who has talked particularly tough on Greece, said Germany will do its utmost toward "making this last chance a success" — provided Greece does its part.
In Athens, Tsipras is widely expected to reshuffle his Cabinet Friday or over the weekend, following a rebellion within his party over a parliamentary vote to approve the measures demanded for the bailout talks to start.
A little more than a quarter of the 149 lawmakers from Tsipras' radical-left Syriza party either voted against or abstained in Wednesday's vote, including two cabinet members as well as the parliament speaker and the former finance minister, Yanis Varoufakis. Tsipras still won an overwhelming majority as three opposition pro-European parties backed the proposals.
The legislation, which includes consumer tax increases and pension cuts, was demanded as a precondition to the launch of negotiations on a third Greek bailout. Elements of the bill are being implemented immediately, with changes to consumer taxes coming into effect Monday, the finance ministry said.