The force is scouring CCTV from the fixture, which Wolves won 3-1, in a bid to trace everyone who had a hand in the trouble.
They said their officers were "confronted by a significant group of home fans" involved in "violent disorder" who had gone out of their way to target police lines.
Port Vale said they would fully back any action against any of their fans that were found to be involved.
The trouble started after the final whistle at Vale Park and spilled out into nearby Burslem town centre where missiles were thrown at officers and a marked police car was damaged.
Staffordshire Police Chief Inspector Dave Bird 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."