Erdogan has complained that the poem was a smear -- and one that violated an obscure German law against insulting representatives of foreign countries.
Whether the comedian will be charged will be decided by a prosecutor. But under this law, no prosecution could have even been considered without the approval of the government.
In a news conference, Merkel acknowledged there had been differences of opinion within her coalition government.
"The result is, that the German government grants the right to prosecute Jan Böhmermann," she said. "I want to explain further that Turkey is a state with whom we have strong ties. There are a lot of Turkish citizen living in Germany. We have strong economic relations and our responsibility to work together in the NATO."
But she said the final decision in the case would rest with prosecutors and the courts, and the government's green light to proceed should not be seen as a prejudgment.
Coalition partner 'Decision is wrong'
Boehmermann faces a maximum sentence of three years in prison.
Thomas Oppermann, the head of the Social Democrats, Merkel's partner in the coalition government, took a different line from Merkel.
"I think this decision is wrong," he tweeted. "Criminal proceedings against satire for 'insulting a majesty' do not fit in with modern Germany."
The decision began to draw international condemnation, as well.
"It's completely insane," said Irish comic writer Graham Linehan. "Edrogan and his gang are just nuts."
The controversy began when Boehmermann, host of the show "Neo Magazin Royale" on the German public TV station ZDF, read on the air a sexually crude poem that referred, jokingly, to Erdogan.
The Turkish Embassy initiated legal action. And several days later, Erdogan -- a proud man not known for his self-deprecating humor -- joined the fray as well, lodging a criminal complaint against Boehmermann.
It's not the first time Erdogan has taken offense. More than 1,800 criminal cases
have been opened in Turkey for insulting the President since he came to power in 2014, according to the country's justice minister. A year ago, two cartoonists in Turkey
were sentenced to 11 months and 20 days in jail for insulting him with a drawing. The sentence was later reduced to a fine.