Grant was out of contract at Gresty Road and makes the short switch to Vale Park on a free transfer, putting pen to paper on a two-year deal.
The 28-year-old becomes Page’s fifth new signing of the summer, following the arrival of Remie Streete, Sam Foley, Sam Kelly and Ben Purkiss.
“Anthony will be a great addition to our squad and will really help strengthen the midfield area,” said Page.
“He has performed very well for Crewe over the last couple of years and we’re delighted to have him on board as we head into the new season.”
Grant was a regular for Crewe since signing on a free transfer from Stevenage in the summer of 2013.
He was runner up in the Alex’s player of the year awards for last season, finishing only behind goalkeeper Ben Garratt.
The former Chelsea trainee started 42 league games during the 2014/15 campaign and was outstanding in his side’s 2-1 win at home to Vale last September.
Grant has also had spells at Southend, Luton, Wycombe and Oldham.