The Alex have made capturing Inman's signature one of their top priorities this summer after he impressed on a loan spell with the reigning Johnstone's Paint Trophy holders last term.
Inman, 21, turned down a fresh deal at St James' Park this summer and made no secret of his desire to engineer a switch to Gresty Road.
However, the Magpies rebuffed Crewe's initial advances and the club feared that, given Inman's age, a tribunal would be needed to dictate his fee.
Yet Steve Davis' side confirmed on Tuesday that Inman had agreed terms to a three-year deal for an undisclosed fee.
Now, with all of the paperwork signed off, the Australian-born midfielder is available for selection against Leyton Orient this weekend.
Speaking to the Alex's official website on Tuesday, Davis said: "He will add strength to our options and will be a real asset to us.
"He has the potential to develop into a very good player and to those pessimistic supporters, who didn't think it would happen, they should have a bit more faith in the club.
"It has been complicated with the negotiations and a long, drawn-out process but Brad wanted to come here and we wanted him here, so we did all we could and had to remain patient."