Supported by his business partner, Norman Smurthwaite, Wildes's Alchemy Investment Group's bid finally went through on Tuesday evening.
The deal had to be done by midnight before the bidders' exclusivity period expired with the club's administrators.
That could have led to another offer being accepted from rival bidders.
Vale had been in administration since March, but the Football League confirmed they are now out.
"I would like to welcome Paul Wildes and Norman Smurthwaite to the Football League and thank them for their efforts in helping to rescue Port Vale Football Club," Football League chairman Greg Clarke said.
It has been a long process but we are delighted to have finally taken over the club and look forward to a long and successful association
"The Football League remains committed to improving the financial health of professional football and, as with any club which has been through a period of administration, will continue to monitor the situation at Port Vale over the coming years to help ensure the club remains on an even financial footing."
Vale's new owners received approval from the Football League last week, but had to wait before all the final legal details were in place.
Sheffield-born Wildes, 35, who also attempted a takeover at Darlington earlier this year, runs a variety of businesses, from hotels to hairdressers.
A former finance director for the Caudwell Group, which owned brands such as Phones 4u, Wildes owns some of the franchise for the hotel chain Holiday Inn.
He set up his own company in 2007, Mansion Group, which he sold three years later, before establishing a new holding company, the Wildes Group.
In a statement on the club's website, Wildes, who was named as the preferred bidder last month after Keith Ryder's summer-long bid ended, said: "I would like to thank the Port Vale supporters, once again, for their ongoing patience and support.
"It has been a long process but we are delighted to have finally taken over the club and look forward to a long and successful association.
"As I have previously stated, our intention is to develop the club both on and off the field."
Micky Adams' side are second in League Two, although they had not won in five league games before beating Bristol Rovers 4-0 on Tuesday night.
Port Vale have exited administration after Wirral businessman Paul Wildes completed his drawn-out £1.25m purchase of the League Two club.