The sides clash in Group D at Red Star Belgrade's stadium on Sunday and Chelsea defender Ivanovic said the outcome could have a telling impact on Serbia's bid to reach their first major tournament since the 2010 World Cup.
Having played against Arnautovic several times in the Premier League, the 32-year-old right back heaped praise on the Stoke City forward, who was born to a Serbian father and an Austrian mother.
"I have the utmost respect for Arnautovic because he has matured in the last year and a half or two," Ivanovic told a news conference.
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"He has drawn a lot of attention here because of his heritage and in footballing terms, he has perhaps improved beyond the quality of the club he plays for," he added.
"We have to close him down and if we do, we have a good chance of keeping a clean sheet."
Wales, Serbia, Austria and Ireland each have four points from two games in what is shaping up to be a fiercely competitive section, with only the group winners guaranteed an automatic berth in the 32-team tournament in Russia.
Georgia and Moldova are yet to earn a point.
Ivanovic headed Serbia's second goal in a 3-0 win at Moldova on Thursday while the Austrians twice came from behind to salvage a 2-2 home draw against Wales thanks to an Arnautovic double.
The towering 27-year old said he would refrain from celebrating if he scored against Serbia and defender Aleksandar Dragovic, born to Serbian parents in Vienna, echoed his view.
"It would be unfair to the fans," Arnautovic told reporters. "It will not be an easy night for me because there will be a lot of emotions, but I will play my heart out."
Dragovic added: "We are playing against my parents' country and most of my family will support Serbia, but I think my grand-parents will cheer for me."
Ivanovic said his team needed to keep up their momentum.
"The match against Austria is incredibly important for both teams as it could determine the entire qualification process," he said.
"In the past few campaigns, we were pretty much out of contention after the opening two or three games whereas the situation is entirely different now."
(Editing by Ed Osmond)
