Dynamo Kiev also advanced to the playoff round with a 2-0 home win over Dutch side AZ Alkmaar, as did Belgians Gent after they beat visitors Rapid Vienna 2-1.
Benfica, who won Europe's premier club competition in 1961 and 1962, saw last season's woes spill over into this term as Zivkovic, who endured four difficult seasons in Portugal, rubbed salt into their wounds.
Having seen the domestic league and cup double go to Porto last term, Benfica missed a string of chances against PAOK in a one-sided first half as they enjoyed 72% of possession.
They were punished after the break as Dimitrios Giannoulis poked in the opener from close range in the 63rd minute after a one-two with Chuba Akpom and substitute Zivkovic made it 2-0 in the 75th with a sweet shot inside the near post.
Rafa Silva pulled one back with a stoppage time header but it came too late to avoid an embarrassing defeat which will pile pressure on manager Jorge Jesus, who returned to Benfica during the close season for a second spell in charge.
Having been in charge from 2009 to 2015, Jesus went on to win the elusive double of the Brazilian league title and Copa Libertadores with Flamengo in 2019 but now faces a mammoth task to get Benfica back to winning ways.
Dynamo soaked up pressure on home soil against Alkmaar in the first half before Luxembourg forward Gerson Rodrigues fired them ahead just after the break with a neat finish into the roof of the net thanks to good work by Vitaliy Buyalskiy.
Mykola Shaparenko sealed their passage with a late header after visiting keeper Marco Bizot misjudged Tomasz Kedziora's cross from the right which left the midfielder with the simple task of nodding the ball into an empty net.
Niklas Dorsch headed Gent into a 36th-minute lead against Rapid from a pinpoint Sven Kums assist and Roman Yaremchuk added the second with a penalty on the hour before Yusuf Demir netted a stoppage time consolation for the visitors.
In Wednesday's standout fixtures, 1991 European Cup winners Red Star Belgrade visit Cypriots Omonia Nicosia while Dinamo Zagreb are at Hungarian champions Ferencvaros. (Writing by Zoran Milosavljevic; Editing by Ken Ferris)