Lajcak was the sole candidate for the position, and was elected by an overwhelming majority.
He will succeed Fiji's former ambassador to the United Nations, Peter Thomson, in September and chair the assembly's 72nd session that month. The post lasts a year.
The sole candidate for the position, Lajcak was elected by an overwhelming majority. He will be the first Slovak in the job.
Under the current rotation system, regional groups in the assembly put forward consensus candidates for president.
Addressing the assembly, the 52-year-old career diplomat set out a series of priorities, including "preventive diplomacy," responding to the global migrant crisis, defending human rights and reducing inequality.
"We can do more to bring the UN closer to the world's citizens," he said.
He also stressed the importance of the Paris climate accord, refusing to comment about an expected withdrawal by the United States.
The issue of climate change "is a priority and a question of survival for a number of UN member states," he said.
"The Paris agreement is one of the biggest successes of the international community lately," he added. "We must implement it."
Although US media on Wednesday reported that President Donald Trump has decided to pull America out of the Paris accord, he is keeping the world guessing, saying only that an announcement would come in the "next few days."
UN Secretary General Antonio Guterres said on Tuesday that it is "absolutely essential" the Paris agreement be implemented.
Lajcak was a contender to succeed Ban Ki-moon as UN Secretary General last year, a post that went to Portugal's Antonio Guterres.
The polyglot became the youngest Slovak ambassador -- and the youngest ambassador posted in Japan -- at the age of 31.
Later, he played a role in stabilizing the former Yugoslavia in the aftermath of the wars that tore the Balkans apart in the 1990s.
He was in charge of organizing a referendum on the independence of Montenegro from Yugoslavia in 2006 before his appointment the following year as the European Union's special representative in Bosnia and Herzegovina.