The insurgents said the bombing was in revenge for a Koran-burning incident at a US air base near Kabul a week ago.
The burning, which the US says was unintentional, has sparked violent protests across the country.
Nato said it had no reports of international forces being among the casualties in Jalalabad.
The airport serves both civilian and international military aircraft. Witnesses reported seeing at least four destroyed cars at the gates.
'Late reaction'
Taliban spokesman Zabiullah Mujahid said in an email to media: "This attack is revenge against those soldiers who burned our Koran.''
President Hamid Karzai has appealed for calm amid anger at the incident at Bagram air base near Kabul.
In his televised address on Sunday, Mr Karzai "condemned with the strongest words" the treatment of the Korans but added: "Now that we have shown our feelings it is time to be calm and peaceful."
But the BBC's Bilal Sarwary in Kabul says Western officials have complained that the Afghan leadership, including Mr Karzai, did not do enough to stop the violence.
Several Western diplomats said they would have preferred the Afghan president deliver a video address swiftly and ask for an immediate end to violence by demonstrators. But they say his late reaction allowed violent protests to continue, according to our correspondent.
Manhunt
Muslims consider the Koran the literal word of God and treat each book with deep reverence.
Various apologies from Nato and the US over the last week failed to cool the anger on the streets.
Meanwhile, Afghan authorities are still hunting a 25-year-old Afghan policeman believed to have shot dead two senior Nato officers at the interior ministry in Kabul on Saturday.
Afghan officials named the suspect as a police intelligence officer, Abdul Saboor, from Parwan province.
The dead Nato officers are believed to have been Americans.
The US, Britain, France and Germany have now withdrawn civilian staff from Afghan institutions. It is unclear how long the withdrawal will last.
