He said Mrs Merkel was by far Europe's most important leader, and that the EU had become a vehicle for Germany.
Mr Trump was giving details of his foreign policy goals in an interview with British and German newspapers.
He told The Times and Bild his priority was to create fairer trade deals for the US and have strong borders.
He said the US had to address its trade deficit with the rest of the world, particularly with China.
The emphasis for his administration should be smart trade, rather than free trade, he said.
Asked about a possible deal with Russia, he said nuclear weapons should be part of it and "reduced very substantially", in return for lifting US sanctions.
Turning to the Middle East, he condemned the decision to invade Iraq in 2003 as possibly the worst decision ever made in the history of the country, and said safe zones should have been created within Syria and paid for by the United States' Gulf allies.
Mr Trump also spoke about the UK and Brexit, saying he thought the UK was "so smart in getting out" of the EU.
"Countries want their own identity and the UK wanted its own identity," he said.
"I think you're doing great, I think it's going great."
He predicted that more countries would follow the same path.
"I think people want . . . their own identity, so if you ask me ... I believe others will leave."
During the interview, held in Mr Trump's offices in Trump Tower, New York, the president-elect said he thought Mrs Merkel was the "by far the most important European leader".
"If you look at the European Union, it's Germany - it's basically a vehicle for Germany," he said.
"I think she made one very catastrophic mistake and that was taking all of these illegals."
He linked the migrant issue with the UK referendum vote to leave the EU.
"I do believe this, if they (EU countries) hadn't been forced to take in all of the refugees, so many, with all the problems that it . . . entails, I think that you wouldn't have a Brexit.
"It probably could have worked out but this was the final straw, this was the final straw that broke the camel's back."
He also stressed that he would "start off trusting both Russian President Vladimir Putin and Mrs Merkel" on taking office, but would "see how long that lasts".
Talking about international security, Mr Trump argued that he had said "a long time ago that Nato had problems".
"One; that it was obsolete because it was designed many many years ago, and number two; that the countries weren't paying what they're supposed to pay."
He said that only five countries were paying what they should into the Nato budget.
"Five. It's not much... With that being said, Nato is very important to me."