BERLIN—German Chancellor Angela Merkel found herself in a public showdown with the sister party to her Christian Democratic Union on Friday, drawing sharp criticism for her opposition to a national cap on newly arriving migrants.

Ms. Merkel spoke in Munich at the annual conference of the Christian Social Union, sister party to the chancellor’s own party. CSU delegates unanimously backed a motion that demanded Berlin limit the number of migrants entering the country.

But Ms. Merkel told the gathering that national responses alone won’t help. Instead, what’s needed is a combination of protecting national borders, seeking European solutions and fighting the root causes of migration and flight, she said.

“With this…we save lives and we will reduce the number of refugees,” Ms. Merkel said. “With this approach to reduce the number of refugees—and in contrast to setting a one-sided upper national limit—we will be able to act in the interest of all.”

She said isolation and a failure to seek solutions aren’t the right response in the 21st century.