A video from a group calling itself The Islamic Army in Iraq - which was reported to have killed two Pakistani captives last month - showed a passport and business card which identified the seized envoy as Fereidoun Jahani, an Iranian embassy official in the Shia Muslim holy city of Kerbala.
Iranian state television said the confirmation had come from its embassy in Baghdad.
The group said in a statement it had "kidnapped the Iranian consul in Karbala because he had been involved in inciting sectarian strife and operating outside the sphere of diplomacy".
The group also warned Iran "against flagrant interference in the affairs of Iraq".
It did not mention any threats against the captive or demands by the group.
The Islamic Army in Iraq had previously captured a Filipino and threatened to kill him, but released him after the Philippines withdrew its forces from Iraq.
The group also claimed the capture of two Pakistanis and an Iraqi in July and threatened to kill them if their company did not pull out of the country. The two Pakistanis were later reported killed, while the Iraqi was freed.
Both Iraqi and US officials are uneasy about attempts by Iran, which presents itself as a centre for the Shia world, to gain influence among neighbouring Iraq's Shia.
Iran's ISNA student news agency quoted an Iraqi police spokesman in Karbala as saying 1200 Iranians had been arrested there in the past 10 days.
He said most of them had been deported across the Iranian border, but 32 had been kept for questioning.
Tens of thousands of Iranians have flocked to the holy cities of Karbala and Najaf on pilgrimage since US-led occupation forces toppled Saddam Hussein last year.
Iraq's defence minister accused Iran last week of "blatant interference" in Iraq's domestic affairs. Iran denies the accusations.