Official results from half of polling stations on Sunday showed Dan garnering about 50.5 per cent of votes, while exit polls published right after voting ended had him at 54-55 per cent.
The soft-spoken mathematician made a last-minute dash to the top in recent days after weeks of trailing George Simion, a hard-right supporter of Trump and a eurosceptic wanting to end military aid for Ukraine in its war with Russia. Simion did not concede.
Dan, 55, had campaigned on a pledge to fight rampant corruption, maintain support for Ukraine - where Romania has played an important logistic role - and to keep the country firmly within the European mainstream.
If final results confirm his victory, Dan would need to nominate a prime minister to negotiate a majority in parliament to reduce Romania's budget deficit - the largest in the EU - as well as to reassure investors and try to avoid a credit rating downgrade.
"There will be a difficult period ahead, necessary for economic rebalancing to lay the foundations of a healthy society. Please have hope and patience," Dan told supporters after exit polls were published.
Simion won 41 per cent of ballots cast in the first round of the election two weeks ago and was the top vote-getter.
The president of Romania, an EU and NATO state, has considerable powers, including being in charge of the defence council that decides on military aid. The president will also have oversight of foreign policy, with the power to veto EU votes that require unanimity.Â
The elections is being closely watched across Europe amid concern over a rise in the popularity of Trump among voters angry over with mainstream elites over migration and cost of living pressures.
Staunchly pro-EU and NATO, Dan said in the run-up to the election that Romania's support for Ukraine was crucial for its own security against a growing Russian threat.
Political analysts had said victory for Simion would have risked isolating Romania abroad, eroding private investment and destabilising NATO's eastern flank.
It also would have meant that Hungary's fiercely anti-immigrant leader Viktor Orban, a long-time Trump ally, and Slovakia's Robert Fico - who both oppose military aid for Ukraine - would gain a new ally in the European Council in decisions on aid for Ukraine, energy, sanctions against Russia or the EU's budget.
Speaking after voting ended, Simion said his election was "clear."
"I won!!! I am the new President of Romania and I am giving back the power to the Romanians!" Simion said on Facebook.
The election took place nearly six months after the initial ballot was cancelled because of alleged Russian interference - denied by Moscow - in favour of far-right frontrunner Calin Georgescu, who was banned from standing again.
