Forty-seven percent of the 1,000 people who took part in the poll by the ARD TV station said they were satisfied with Merkel's work, 12 points down from her July rating.
It was Merkel's second worst poll score since she was re-elected in 2013. In February 2016, 46 percent of respondents gave favourable opinions of the leader.
In this latest poll, just 34 percent said they were satisfied or very satisfied with the government's welcoming policy towards refugees.
It was the lowest result on the issue since it was first included in the monthly poll in October.
Some 65 percent said they were dissatisfied with the policy.
Horst Seehofer, the conservative premier of Bavaria and a staunch critic of the government's refugee policy despite being one of Merkel's allies, saw his popularity rating soar 11 points to 44 percent.
Germany's first attacks claimed by the group Isis took place in July: a 17-year-old Afghan refugee attacked a family on a regional train near Würzburg with an axe, and later a 27-year-old Syrian refugee blew himself up in Ansbach.
A few days after the attacks, Merkel continued to defend her migration policy, repeating the "We can do it" slogan she first used in the summer of 2015. That year, Germany welcomed more than a million migrants.
A separate poll on Wednesday showed however that nearly 70 percent of German respondents did not blame the attacks on the Chancellor.
Merkel is expected to run for a fourth term in the next general election, due to take place in just over a year.
In September, regional elections will be held on her home turf of Western Mecklenburg-Pomerania, where her party is trailing in the polls.