Komorowski, 58, did not formally claim victory in an address to a crowd of party members in Warsaw, the Polish capital, saying the runoff election votes were still being counted.
But he expressed optimism that he would be the next president.
"Tonight we open a small bottle of champagne and tomorrow we will open a big bottle," he said.
Exit polls have a small margin of error, and official results are not expected until Monday.
Kaczynski concedes defeat
Kaczynski, surrounded by supporters chanting "Jaroslaw!", conceded defeat by congratulating Komorowski.
"I congratulate the winner, I congratulate Bronislaw Komorowski,'' Kaczynski said.
Komorowski pledged that as president he would serve all Poles and recalled that the voting was forced by national tragedy.
"We thank everybody - the more so that it was an unusual campaign, a difficult campaign held in the shadow of catastrophe," Komorowski said.
An election was originally set for the autumn but had to be called early to replace Lech Kaczynski.
The air crash was the worst tragedy to strike Poland in decades and set the tone for an election campaign free of the dirty political manoeuvring that often precedes the country's elections.
Both presidential candidates were former anti-communist activists with conservative, Roman Catholic upbringings, yet they split on crucial issues, primarily the role of the state in the economy.
The vote will help decide the speed and scale of economic overhaul and set the tone for Poland's ties with its European Union partners and with Russia.
Komorowski would be expected to smooth the way for the government to continue privatising state-run companies and cut welfare benefits, if the exit poll results are upheld.
In Poland, the government led by the prime minister sets policy, but the president appoints many senior officials, has a say in foreign and security policy, and can propose and veto laws.