Officers arrested 23 people and Staffordshire Police said it was "confronted by a significant group of home fans" involved in "violent disorder" who had gone out of their way to target police lines.
The force is now scouring CCTV from the day's fixture, which Wolves won 3-1, in a bid to trace everybody who had a hand in causing trouble.
Port Vale said they would fully back any police action against any of their fans involved.
After the final whistle, the trouble spilled over into nearby Burslem town centre in Stoke-on-Trent, where missiles were thrown at officers and a marked police car was damaged extensively.
There were reports that innocent bystanders were also caught up in the melee.
Two adults also suffered dog bites from police dog units and were treated by paramedics, according to the force.
Among the arrests; nine people were detained before kick-off while another four were arrested during the match for offences including setting off flares and going onto the pitch.
Police said they were "disappointed" a "significant number" of fans had decided to cause trouble at the fixture despite lengthy preparation between club officials and officers aimed at preventing problems.
The force added fans from both sides had been involved but praised a group of Wolves fans who "co-operated with police by waiting in the away coach park whilst police worked to clear the area".
Chief Inspector Dave Bird, of Staffordshire Police, said: "We are disappointed by the actions of this significant number of so-called fans who attended (Saturday's) fixture intent on becoming involved in disorder.
"All those arrested will be questioned about their actions and CCTV evidence from (Saturday) will be examined by officers.
"The behaviour of this significant minority doesn't represent the club or our city.
"We will now continue to work with Port Vale to reduce the opportunity for future disorder."
Bill Lodey, Port Vale club secretary, said: "The club are bitterly disappointed regarding the violence that has been reported in Burslem town centre.
"We will fully support any actions taken by the police in apprehending those responsible and they can expect severe penalties."