Yushchenko told reporters that "what we did during the last 30 days was a tribute to our ancestors."
Yanukovych said he had voted "for the future of the Ukrainian people."
Central Election Commission Chairman Yaroslav Davydovych said all polling stations had opened on schedule and that voting is proceeding normally.
Sandra Gayle, an election monitor from the Organization for Security and Cooperation in Europe (OSCE), also told reporters the balloting has so far been smooth.
Gayle: "It's going very well. We haven't found any violations. It's all going to plan. We're very happy with it. And everyone's very excited about the election."
Reporter: "Do you think it will be much better this time?"
Gayle: "I think so, yes."
Some 12,000 foreign observers are monitoring the voting.
Yanukovych officially beat Yushchenko in their first runoff on 21 November, but the Supreme Court later ruled the vote invalid.
First results are not expected until tomorrow.
[For more RFE/RL coverage and analysis of the political crisis around the presidential election in Ukraine, click here.]