In a videotape broadcast on Al-Jazeera television, Zawahiri hailed Zarqawi - who died in a US air strike on 7 June - as "a hero" and "the prince of martyrs".
Speaking to a backdrop of a picture of Zarqawi, Zawahiri addressed US President George W. Bush, saying "try in vain to dream of security."
"Yes O Bush, none of us is killed without us avenging him, with the help of God.
"As to the Americans … tell me how many of you really died (in Iraq), how much your economy lost and how the morale of your soldiers is crumbling.
"Bush is lying to you when he says that you will be victorious if you kill Osama bin Laden, (the leader of Afghanistan's Taliban movement) Mullah Omar and the members of Al-Qaeda and the Taliban.”
The videotape was televised as twelve people were killed and 20 wounded when a bomb went off outside a Sunni mosque in the village of Hibhib, north of Baghdad.
In the southern city of Basra, at least 10 people were killed and 18 wounded when a suicide bomber blew up a vehicle near a petrol garage.
Meanwhile a daytime gunbattle in Baghdad between Sunni insurgents and Shiite militiamen has prompted the Iraqi government to tighten the Baghdad curfew.
An interior ministry official said the street fighting on Haifa Street on Baghdad's west side broke out after militiamen loyal to Shiite radical cleric Moqtada al-Sadr's Mehdi Army came under attack from Sunni Arab gunmen reportedly carrying rocket-propelled grenades.
The militiamen were escorting Shiite worshippers to special weekly prayers at the Baratha mosque, called by Sadr to commemorate the 11 people killed in a suicide bombing in the mosque last week.
The militiamen retaliated and five of them were killed in the ensuing battle.
The violence came despite a massive nine-day-old security operation involving 60,000 Iraqi and US personnel dubbed Operation Forward Together.
Authorities have also found the bullet-riddled bodies of five government employees who were part of a group of more than 60 staff abducted from state-owned factories on Wednesday.
Masked gunmen seized the staff as they were heading home from work in Taji, on Baghdad's northern outskirts.
Sunni Arabs, who made up nearly half of the hostages, were swiftly released, highlighting the sectarian motivation of the kidnappers.