The German-born rich-lister moved to New Zealand in 2010 and has been pursued by the US government on a range of indictments.
Mr Dotcom made his fortune from Megaupload, a file-sharing website often used to share pirated movies and music.
Mr Goldsmith said he had "received extensive advice from the Ministry of Justice on this matter", The New Zealand Herald reported.
"I considered all of the information carefully, and have decided that Mr Dotcom should be surrendered to the US to face trial," he said.
"As is common practice, I have allowed Mr Dotcom a short period of time to consider and take advice on my decision. I will not, therefore, be commenting further at this stage," he said.
In a post to social media, Mr Dotcom said Mr Goldsmith’s decision showed New Zealand was an "obedient US colony in the South Pacific".
He maintained that had no control of what Megaupload’s users decided to upload on the website.
