The ruling Congress will take a third position with 9-17 seats, says the survey.
"The last minute jitters may have kept the voters away from a new party like ours," said AAP member Yogendra Yadav.
However, political analysts say AAP has played a major role in the election this year.
"AAP also needs to be complimented for having energised campaigning in Delhi which were otherwise very lackluster. They introduced innovative methods of campaigning and revived a moribund BJP forcing them to take remedial measures," said senior journalist and columnist Swapan Dasgupta.
"They actually helped BJP in getting maximum of the anti-incumbency votes as it pulled out people to vote in record numbers," he said.
"We made our own calculations and we expect to get a clear majority. But it is not a poll count yet. Nevertheless, we hope to form the government," said BJP spokesperson Meenakshi Lekhi.
The Congress refused to comment on exit polls results and said Sheila Dikshit would deliver again.
"The 9-17 figures for the Congress is not correct. Harsh Vardhan is a nice man but Sheila Dikshit is much experienced and a tall leader," said Congress leader RK Anand.
He, however, denied that the Congress leadership left Dikshit high and dry by not campaigning adequately for her. "Congress Union ministers were too busy with their ministerial work or weere in their own constituency. This happens when your party has a government both at the Centre and the state," Anand said.
Unveiling the strategy how the BJP used Narendra Modi, Lekhi said, "Modi's rallies happened in areas where BJP never won."