Saleh said: "The angle that al-Qaeda had a role in [the attack] is very clear."
However, the Taliban has said it was responsible for the attack on April 27.
The "mastermind" of the attack was also involved in an attack on a five-star hotel in Kabul, the capital, in January, Saleh said.
The attack killed at least eight people, including three foreign nationals.
The man, identified as Homayun, was killed in a raid by security forces in Kabul last week.
Another fighter, a woman and a child were also killed in the operation on Wednesday, in which suspected Taliban fighters shot dead three Afghan intelligence agents.
Saleh said the woman was a would-be suicide bomber.
The "cell" had been involved in a series of suicide attacks, car bombings, and assassination attempts and was linked to pro-Taliban fighters based in Miranshah in Pakistan, he said.
The attempted assassination of Karzai was one of the most brazen attacks in recent months, made even more significant by men suspected of links to the government in Kabul.