In an online recording, an Abdallah Azzam brigades leader also warned the group would stage new operations against Hezbollah in Lebanon, in revenge for the Shiite movement's support for Syria's President Bashar al-Assad.
"We announce to the Sunnis in the Levant and in general, and in Lebanon in particular, that Hezbollah military commander Hassan Lakiss was not assassinated by the Jews (Israel)... God himself killed him, through the two martyrs in their attack on the Iranian embassy," Sheikh Sirajeddin Zureikat said.
He was referring to a twin suicide attack on November 19 that targeted the Iranian embassy in the southern suburbs of Beirut, a stronghold of Hezbollah.
On December 4, Hezbollah said Lakiss was killed near Beirut. The group blamed Israel for the assassination.
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The online video does not show Zureikat's face. Instead, it shows the pictures of the two men who carried out the suicide attacks that killed 25 people.
Beneath their pictures, in which they are shown wearing military gear, runs a caption reading: "The martyr Abu Oweiss al-Sidawi (Moein Abu Dahr) God welcome him."
Immediately after the attack on the Iranian embassy, the Abdallah Azzam brigades had claimed responsibility for the blasts.
In the latest recording, Zureikat meanwhile said the attacks were in revenge for "Iran's crimes" in Syria against Sunnis, as well as for Tehran-backed Hezbollah's "crimes".
Hezbollah and Iran back Assad, and as the war in Syria has raged on, Al-Qaeda-affiliated and other radical Sunni groups have joined the anti-Damascus rebellion.
In Lebanon, which until 2005 was dominated by Damascus, sectarian tensions have soared in recent months, pitting Sunnis against Shiites.
"Our operations will come to an end in Lebanon only if two conditions are fulfilled: first, Hezbollah must withdraw its troops from Syria... And secondly Sunni youths must be freed from Lebanon's oppressive jails," Zureikat said.
Dozens of Sunni fighters are being held in Lebanese jails since a 2007 battle in Nahr al-Bared, northern Lebanon, that pitted the Lebanese army against radical Islamists.