Ferrer, looking to bounce back from an opening-round exit in Beijing last week, won his first match of the tournament after a bye in the first round.
"Every week is different,'' Ferrer said. "I played a very good match today, a lot better than last week.''
Ferrer won the first set with a crosscourt forehand and looked set for an easy win, but Levine bounced back, breaking Ferrer in the first game of the second set. Ferrer took control in the third set, breaking Levine twice before winning with his fifth ace of the match.
Ferrer defeated Richard Gasquet of France in last year's final.
"It's very special here,'' Ferrer said. "I have a lot of fans here and winning here last year was a great experience for me.''
The Spaniard will meet Japanese wildcard Takao Suzuki in the third round. Suzuki beat Simone Bolelli of Italy 7-6 (5), 4-6, 7-6 (5).
Roddick was coming off a win in the China Open on Sunday, and he said that affected his game against his Czech opponent.
"I feel like I played OK,'' the American said. "I'm tired coming off last week, I've played a lot of tennis. He played well and I'm just happy to be through and play again.''
Roddick, who also had a first-round bye, will face Jurgen Melzer of Austria in the third round. Melzer beat Simon Stadler of Germany 6-2, 6-3.
Roddick trailed 4-1 in the second set but fought back to take a 6-5 lead only to be broken in the final game by Minar, who then won the second tiebreaker.
"You always expect a tough match but not necessarily three tiebreakers and 2 hours and 45 minutes,'' Roddick said. "I was facing new conditions, I was run down from last week and I had a long flight from Spain the week before.''
In the third set, Roddick was given a code violation for an audible obscenity.
"We didn't see eye-to-eye on a couple of things,'' Roddick said. "I probably should have got a warning earlier in the match but I didn't feel I deserved it when I did get it. At that point, I'm not prone to logical thinking all the time.''
Roddick won the third tiebreaker with an ace, one of 37 in the match for the American.
Japanese teenager Kei Nishikori defeated Guillermo Garcia Lopez of Spain 6-4, 6-4 and will take on Gasquet in the third round. The fourth-seeded Gasquet coasted to a 6-4, 6-2 win over Martin Slanar of Austria.
Also, Rainer Schuettler of Germany defeated Lee Hyung-taik of South Korea 6-3, 6-7 (3), 6-3. Schuettler will next face Mikhail Youzhny of Russia, who defeated Flavio Cipolla of Italy 6-2, 6-2.
Agüero: I Was Inspired
Ferrer advances at Japan Open
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Sela upsets No. 1 Ferrer; Roddick advances
