Shami Chakrabarti said Richard O’Dwyer, 23, and his family were “swept up in the injustice of instant extradition”.
Home Secretary Theresa May approved Mr O’Dwyer’s extradition on 13 March.
The US authorities say Mr O’Dwyer’s TVShack website hosted links to pirated films and television programmes.
Lawyers representing Mr O’Dwyer, a student at Sheffield Hallam University, have lodged an appeal against his extradition.
The case was brought by the US Immigration and Customs Enforcement agency, which claims that Mr O’Dwyer’s website earned more than $230,000 (£147,000) in advertising revenue before US authorities obtained a warrant and seized the domain name in June 2010.
A Home Office spokesman said the Home Secretary had “carefully considered all relevant matters” before signing the order to approve Mr O’Dwyer’s extradition.
Ms Chakrabarti said Mr O’Dwyer and his family faced a “nightmare” as a result of the extradition order.
“No-one should be sent anywhere without a basic case tested in a local court and our judges should have discretion to try people at home, especially when they never left their homes, let alone this country, in the first place,” she said.
Ms Chakrabarti will host an event at Sheffield Hallam University’s City Campus later aimed at highlighting Mr O’Dwyer’s case.
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