Bychenko, 26, performed to music from the Swing Kids soundtrack by James Horner and recorded a result of 62.44 points.
Bychenko, who is based in Hackensack, New Jersey, crashed to the ice once during the short program, but still recorded a score high enough to book his place in the second day of competition.
Short track speed skater Vladislav Bykanov failed to reach the quarterfinals of the 1,000-meter event on Thursday, finishing his heat in third place in a time of 1:27.796 minutes.
Bykanov needed to finish among the top two to advance to the quarters, but was beaten by over a second by Russia’s Vladimir Grigorev and Tianyu Han of China.
The 24-year-old, who moved to Israel from Ukraine in 1994, also missed out on a place in the 1,500m semifinals by 76 hundredths of a second on Monday.
Bykanov, who carried the Israel flag in the opening ceremony of the Winter Olympics last Friday, will have one more opportunity to make his mark in Sochi when he participates in the 500m competition on Tuesday.
The fifth and final Israeli to compete in the Olympics will be skier Virgile Vandeput. He will take part in the giant slalom on Wednesday before also racing in the slalom next Saturday, the day before the closing ceremony.
The first Israelis to return from Sochi will be Evgeni Krasnopolski and Andrea Davidovich, who were delighted at finishing the pairs figure skating competition in 15th place on Wednesday.
“There was a lot of pressure with this being the Olympics, but we met the expectations by qualifying for the free skating,” said Krasnopolski.
“We could have finished slightly higher, but considering these were our first Olympics, we are very pleased.”