The Ukrainian Premier League restarted on Tuesday on the country's Day of National Flag, another show of resistance following Russia's invasion in February which brought a premature end to last season's championship.
Zinchenko, who has represented Ukraine 52 times, joined Arsenal from Premier League champions Manchester City in July.
"It's so important to show the rest of the world that the Ukrainian people don't give up, and in any situation they carry on. It's all about our people," Zinchenko said in an interview on Arsenal's website.
Although only certain venues with bomb shelters are being used for UPL games and no fans are allowed into the stadiums, Zinchenko said people needed football.
"I'd like to say a massive thanks to the Ukrainian federation, and president (Volodymyr) Zelenskiy who said we needed to play our competition in Ukraine," he added.
"I know it's not going to have the best facilities but we need to keep going and that's the most important thing for all Ukrainians. I can't wait and I'll watch all the games if it's possible."
