The Tehrik-e-Taliban maintains that Mehsud had left the site more than 40 minutes before the attack and was safe and well.
Abbas said: "We have kicked them out of their base in South Waziristan and there is a complete disconnect from the various sections of this organisation.
"They have been demoralised, partly dismantled, partly defeated, and in great disarray, so this a great success."
The use of drone attacks in Pakistan is controversial.
They have killed more innocent people than Taliban fighters, but the US believes it is an effective way of targeting people hiding out in the remote areas of the Pakistan-Afghanistan border.
Kamal Hyder, Al Jazeera's correspondent in Islamabad, said: "There have been reports of Mehsud's death for two weeks now.
"Then, an audiotape came out in which he denied he had been killed. Then there were reports that he was wounded.
"Today, state TV reported that he had been killed. Since then, the report has gone off air.
Imtiaz Gul, a Pakistani security analyst, told Al Jazeera: "Based on the information that we have been getting since January 14, when the drone struck, we can say that the confirmation of Mehsud's death was just a matter a time."
Baitullah Mehsud, the former leader of the Pakistani Taliban, died last August but it took the Taliban a number of weeks to admit that he had been hit in the missile strike which killed him.
Hakimullah issued an audiotape saying he was well. But he gave no indication when the message was recorded.
The Taliban released another tape on January 17, hoping no doubt to quell the rumours.
In the tape, Mehsud said: "Today is the 16th of January. I want to confirm to all my mujahidin brothers that I am Hakimullah and I am alive and in good health thanks be to God and that I was not injured in the rocket attack."