Cluj - who held the Scottish champions to a 1-1 draw in the first leg of the third-round qualifier - prevailed 5-4 on aggregate.
Celtic lost to AEK Athens at the same stage of the competition last season. They'll now face either Sheriff Tiraspol or AIK in the Europa League's playoff round.
"It's suicidal the goals we gave away tonight," manager Neil Lennon told BBC Radio Scotland.
Cluj took the lead when Ciprian Deac's header found the back of the net in the 27th minute.
Ryan Christie responded in the 76th minute with a sweeping finish from close range that sent Celtic Park into delirium. But the Romanians fought back again, with Omrani scoring his sixth goal of the qualifying campaign to give his team the advantage on away goals.