With 91.21 percent of ballots counted, Fernandez has 47.83 percent of the vote, Macri has 40.66 percent.
"I want to congratulate President-elect Alberto Fernandez," Macri said, addressing the citizens.
He added that he had invited Fernandez to a meeting on Monday.
According to Argentine legislation, the victory in the first round is secured for the candidate who will gain either more than 45 percent of the vote, or more than 40 percent with a 10 percentage points lead over the closest opponent.
Besides the presidential election Argentina held the parliamentary voting on Sunday. The Argentine voters had to elect 130 out of 257 members of the Chamber of Deputies, 24 out of 72 members of the Senate as well as four governors.